Hi! Why're you here?
You could be examining this guide for a number of reasons. You could've stumbled upon a boss you're unsure about in light of murder technique in the story, or you've attempted Time Attack enough times without achieving the 'Bump Master' achievement, or perhaps you're here simply because you've spotted the 'Time Attack' option in the menu and are looking to scour up some intel about what, exactly, it is.
The latter option is the easiest to answer, so I'll start with that.
Incidentally, those of you here for mere curiosity reasons, just don't kill any household pets and you'll be fine.

If you're not here for Time Attack, you can simply skip to the boss of your choice with the contents tabs on the right.

Simply put, Time Attack is a mode in which you battle all bosses in Ys I one after the other.
Don't worry though; you regain your health after beating each one. So basically it's a Boss Rush mode where you regenerate health between battles. Simple, no? Well, mostly.

To unlock Time Attack you need to have beaten the story of Ys I at least once. I don't believe it matters what mode or difficulty you complete it in, just stab Dark Fact's face until he collapses into a pile of empty cape and fail and Time Attack will be yours.
And then what can you expect? Nothing unfamiliar, surely.

Out of the seven instances of Time Attack, this one's the one I generally have the most trouble with, with Dark Fact as a close second. Why? Because I get overambitious and charge into the flames unnecessarily. As with most bosses, your primary asset to beat this green-robed menace is patience.
He'll be teleporting in and out, to briefly stand still and summon a flame between his hands, which apparently signifies he's summoning the flames from the walls, too.
The flames from the walls are the only thing that can hurt you in this fight. The bad news? They're tricky to maneuver and will often block your route to Jenocres' possibly fleshy manifestation in which you'll be wanting to stick your sword.

I haven't found any particular technique that feels very technique-y for this fight, but with a little practice and a moderate amount of patience, waiting to strike at the right moment, this should be easy enough.

You shouldn't have any trouble with him at all in the campaign: Even on Nightmare, you're almost certain to be overleveled and able to take him down very easily. (If you are having trouble in the campaign, you are advised to level yourself up one and try again)

This guy's probably my favourite boss, because bouncing around hacking at his segments whilst he helplessly circles around me trying to get his jaws in just looks too cute, and it's really easy to control him.
And that's how you'll beat him: Control.
This guy's only hope of defeating you is stunlocking you in front of his face as he repeatedly bites at your belly, so his grand scheme is limited to: CHARGE RIGHT AT YA.
Fortunately he hasn't got the agility to get at you when you're behind his head, near his segments, which is where you can hit him. Repeatedly.

He'll be circling around and around trying to reach you, but with a little calculation you can stay within his blind spot, hacking at his spines repeatedly. Just don't get too close to his head, or he'll be able to push you out.

Easily done, even on Nightmare mode.

And here's probably a main cause of headaches amongs Ys players. A sudden spike of cruel and annoying hatefulness breaking the so-far smooth difficulty curve of the campaign to tiny, sharp bits of 'argh'.
Said by some to be the hardest boss in the game, this guy has a high chance of being the reason you're reading this very guide.

Thing is though, in Time Attack, he's actually pretty easy. But possibly only because you've figured out how to beat him by the time you're trying Time Attack. In the campaign? He's monstrous.

So here's a few factoids for in the campaign:

• Patience, as always, is your best friend.
  
• More so than patience, you'll be wanting the silver sword. Without it you'll barely do any damage at all, or even none. Don't have the silver sword by this point? Back out of the bat's cavern and start distributing some of the other silver items you've found.*
  
• Especially on Nightmare, you're going to want to be level 10 at this point. Anything less and you'll be a smear on the wall quite quickly.
  
• Taking breaks between every few attempts is a good idea: It'll boost concentration and lessen the chances of a rage-induced popped artery.

*If you've been looking for the silver sword for a long while and still haven't found out where to get it, the following spoiler will tell you where it can be found:You need to have found both the Roda Tree Seed and the Silver Harmonica in the abandoned mines. Eat the fruit so you can talk to ancient trees, return the silver harmonica to Reah so her song wakes those trees, then go walk up to those trees: The eastern one should cough up the silver sword.

Now, how to actually go about murdering this Vlad-ullion character. His cycle is simple as can be. You can quite easily observe his actions: Become tons of bats, chase you around, re-form just long enough to smirk at you at random intervals, repeat.
Beating him, sadly, is not quite as simple and requires some degree of luck.
The general approach is as follows: When he is bats, run around the room in an attempt to both group the bats together and not get hit. Staying closer to the bats is liable to group them together faster, as mostly they each seperately make a beeline to your location, with some degree of randomosity. Staying close is also handy for the following reason: When Vagullion reforms, he is only vulnerable for a very short time, and that time is partially whilst he's still re-forming. If you're too far, or react too late, you'll miss it, and will practically be sitting on his lap as he explodes into a horde of deadly bats again.
Not good for your health.

This guy's a tough one, and you can take comfort in the fact that most everyone thinks so. Though it looks as if it should be possible to beat him without getting hit, I have never witnessed such an event. Be sure to alter your circling routes to the situation and don't get too stuck in the routine. Stay sharp, and you'll get him eventually.

This guy shouldn't give you too much trouble, no matter the difficulty.
Only the circular boomeranging slice-things he throws about can hurt you. There's only ever three on the screen at once, and if you can keep track of them and only attack his body whilst they're way out, he should go down easily enough.
Be sure to run out far after attacking so the next slices he shoots also go far, giving you extra time to run in and slash; they're being shot at you, after all.

On easier difficulties, and on Time Attack, you should be able to hit him more than once every time the blades are out. On Nightmare mode, note that he shoots his next blade as soon as a previous one returns, so you have less time to hang about his vulnerable self than you'd normally have.
Either way, not too much of a big deal. Enjoyable even, I'd say.

Though most people would call Vagullion unfair, I'd say this one's a definite nominee for that title also. Not because he's hard, but because he's absolutely ruthless and very unforgiving to anyone who doesn't know the trick. The trick, I find, is rather unintuitive and frankly kind of lame. It's an interesting concept to have to unlearn the premise of 'try not to get hurt' for this battle, but the result feels exceptionally silly to me.

What this trick may be, you ask? I'm getting to that.
First thing you should know, is something that rather surprised me when I first realised it, is that it's not the giant glowy orb in the centre that you're trying to hit. Rather, it's the larger rocks directly circling it that you're trying to hurt. With a sword. I know, right?
Now the trick is to forego any idea of 'ouch', charge right in from the very start, and circle the central orb along with the greater rocks, dealing constant damage as you do so. You'll get hit too, a lot, but if you manage to keep in there consistently you should take him down fairly quickly whilst you're left with, oh, say just under half your health. This varies per difficulty, of course, but the technique remains the same.
If you're having trouble figuring out exactly how to do this, you might want to skip down to the video section below.

Demon beasties! I love this boss. But don't be fooled by my enthusiasm: It's not easy.
Well, certainly not on Nightmare difficulty, anyway.

The trick with these guys is, only one of them is vulnerable. It's the one that looks the least evil of the two: The one-horned yellow guy whose satellites aren't aflame. And every time you hit him, the two switch places! How inconsiderate.
Also, the only things that can hurt you in this fight are the satellites (the spinning crystals and fireballs), so don't be afraid of the heads themselves.

Theoretically, someone with mathematical prowess and quick reflexes could beat these guys without getting hit even once, and that's definitely what you should be trying. You probably won't manage it, but every time you hit the vulnerable demon without getting hit is more health saved, which means more not dying.
I suppose that'll all fairly straightforward.

There's no real strategy or technique I can impart at this point.* Just try to time your attacks carefully, try not to get stuck between demons and walls, and remember that you can hide behind the wall at the very bottom of the screen if you need time to sit and think.

*Edit: Save for this!
User Nebuchadnezzar kindly commented:
"Yogleks & Omulgur are pretty much easy (on hard) if you know the following trick.
In the beginning of the fight run straight to the bottom wall and wait there at the bottom center of the screen. When the heads are moving towards you - run in a straight line up to the spot where they will intersect to hit them and then back to your position. That way you will hit them multiple times and deal more damage than you get."

This fight really does scale to the difficulty, as they get a lot faster on Nightmare.
Keep calm, and good luck!

PROJECTILES EVERYWHERE!
RUN FOR YOUR HINEY!

For those playing the campaign, if you're having trouble with:
- reaching him: Can't open the door? Be sure to have and equip the amulet given to you not by the old man, but byLuta Gemma on the 11th floor. You may first have to use the monocle to read the Book of Gemma (one of the Books of Ys)
- hurting him: Hint: Silver's a big deal. You'll need to equip all your silver equipment for the best effect.

This is the big one, the bad one, the evil final one, and probably the one where you'll be killing yourself a lot of the time rather than being killed.
Dark Fact, whatever he is, is obviously a lover of general chaos and mayhem, as that is exactly what this fight feels like. Not to mention he's responsible for a lot of the trouble you spend your time clearing up during the story.
What he does is, he randomly bounces around the screen on a seemingly fixed diagonal trajectory, with two huge occult symbols circling the platform constantly spewing fireball-esque bullets at you that on touchdown split into projectiles that fly in eight different directions and- Yeesh, this horned gennelman's put a lot of effort into making you die a flashy death, hasn't he?
Unfortunately the mechanic that makes parts of the floor crumble away at locations where you manage to hit him often means you'll get yourself stuck or just plain tumble into the abyss - Not so much flashy as it is lame. But exciting!

Now so far as strategy goes, unless you're some kind of ninja who can keep track of everything going on the screen at once and calculate the outcome of your choice of future location to an optimum measure, your best bet is probably to ignore the projectiles coming at you and focus on trying to hit Dark Fact.
And not just hit Dark Fact anywhere! Since the floor crumbles at the places you encounter him, you'll be wanting - especially to start with - to hit him near or preferably at the edges of the field. If possible, try to predict his movement enough to leave spaces where you can hit him later. Don't try too hard though, or you'll end up just missing him.

To sum it up, your priorities should be, in order, as follows:

• Damage Dark Fact, lots, quickly.
  
• Don't get yourself stuck
  
• Don't stop moving, and keep changing direction!
  Just because you should ignore the projectiles doesn't mean they're not there. Changing directions when you can will greatly increase your survivability.

Dark Fact is not a friendly bugger, no sir. He will be tough to defeat, and you will likely suffer plenty of injury even on your best run. Also, you will need luck for this one. Keep at it, and prevail!

There's only so much one can convey in text, and some of you might feel this guide was lacking on details, or needs fleshing out on particular points. For those reasons, I will include these videos, submitted to the tubes by one named Vercsase, whose recordings of his efforts in the Nightmare campaign have also helped me when I was uncertain on how to proceed.
Perhaps they can help you also.

Keep strong!