Spoilers, but the Twin Princes are the penultimate boss fight, and the result of a royal family inbreeding for years and then, when that failed to produce a good enough heir, performing terrible and unspecified experiments on their children. So they seem well balanced, as you can imagine. This can be one of the more frustrating fights of the game, but you've come this far, right?
1. The Approach
The fog door for the Twin Princes is miles away from the nearest bonfire. You need to have opened up the last shortcut at the Grand Archives, which opens up the lift from the bottom of the archives all the way to the top. From the top of the lift go right, which takes you outside onto the walkway. The entrance to the boss area is in the distant building opposite, which requires you to run through a bunch of hollows, up some stairs, and then through a bunch of knights. You can either fight all of them or opt to run through. If you're doing the latter then stay roughly in the centre while going through the barricades, then hard right up the stairs to avoid attacks from the knights (although you'll still need to do a lot of dodge rolling).
Once you've got to the top of the steps make a hard right to another lift. This will take you all the way down to the chapel just before the Dragonslayer Armor bonfire. Opening this lift up bypasses all the enemies on the bridge, which makes getting to the boss fight less depressing after each death.
In the first phase of the fight you'll mostly be taking physical and fire damage, before moving into the second phase which adds a lot of magic and some miracles into the mix. The older prince Lorian, the big fella, is more susceptible to lightning and frost damage (and due to the fiery nature of his attack regular pyromancies are not as effective as dark pyromancies, unless they're the higher tier pyromancies), while his younger brother Lothric is a frail thing weak to physical damage. When you first step into the boss area Lorian attacks immediately so you won't have time to apply buffs then; you have to do it before you go in.
2. The Twin Princes' first phase
In the first phase you fight Lorian on his own, who seems to be the heavy for his brother. Lorian, aided by his brother, can teleport over the boss arena in a beam of white light. Usually when he does this he appears behind you and attacks, so when you see him teleport you should move with some urgency. Lorian will perform a teleport attack immediately when you step into the arena, but instead of running towards him wait at the fog door - he'll teleport and appear back in the same place. After this he'll start crawling towards you very slowly, so for ranged builds you can get a load of cheap shots in here very early on.
Occasionally the teleport will take him off screen, before he appears further away from you. At this point Lorian will point his sword in the air and charge it, before slamming it down and shooting a line of magic across the floor. This has long range but is narrow, so it's easy to dodge to the side of it if you're paying attention.
Lorian's attacks are otherwise what you'd expect from a greatsword. They're wide swings, overarm slams, and occasional jabs forwards, but chain together and are performed quickly. Lorian will occasionally stand up to perform a heavy slam (which is very obviously telegraphed and easily dodged). He also has an attack where he scrapes the sword across the floor and into an uppercut slash, which can knock you back and stun you if it hits, as well as sending out a blast of fire. These attacks are both easy to avoid at range and, together with the magic sword slam, present a window to attack Lorian afterwards for both ranged and melee types. Ranged players will have a much easier fight in this stage. Lorian crawls slowly so you can often attack safely while he's making his way towards you, as long as you maintain enough distance to outrange his sword, 'cos that bad boy hurts.
Melee players have two options. You can bait attacks from him at mid range and then either doge through or run in to attack, but move out again quickly and play cautiously to avoid taking any damage. The other option is to go on the offensive. Lorian can't be stagger locked, but he can be stunned if you chain enough damage on him. Doing this is a gamble, since it may mean you use estus flasks that would be useful in the second phase. On the other hand, if you take him down quickly enough you might not need to use them anyway. It's a gamble that only you know if you should make.
Getting through Lorian on his own isn't too bad. It's what happens afterwards that's the bigger problem, and you need to keep as much estus as possible for the second phase.
3. The Twin Princes' second phase
After you kill Lorian an incredibly creepy cutscene will play in which Lothric, the younger of the two princes, will resurrect his brother and climb onto his back, hanging there like a ragdoll and occasionally casting spells to make your life more difficult. The spells do an absurd amount of damage, and if you kill Lorian then Lothric will just to res him again (to approximately half health) until you kill him too. The very tenuous good news is that if you can kill Lothric first then his brother dies automatically, and that Lorian hasn't changed his moveset (although he does do more damage and will teleport more often now so... good news negated).
Lothric has two spells, aside from the resurrection of his brother. One casts homing orbs that track you and do more damage than they have any right to. If a bunch of these catch you at the same time as a hit from the sword it can mean the end of you, so it's best to back off the attack when Lothric casts this, and concentrate on dodging the orbs. He also has a soul dart type projectile, which is easy to dodge if you see it coming but does a tremendous amount of damage if you're not prepared. Lothric will always cast this after Lorian does his charged-sword-magic-slam attack, and often casts it after a teleport.
Luckily Lothric doesn't cast continuously. There are large gaps between his spells where you can concentrate on getting attacks in. If you're able to attack the Princes from behind you can damage both of them at once, which means you have a chance to kill Lothric before he's able to resurrect his brother and go on the attack again. This is much harder for ranged players, but on the other hand ranged players will be able to avoid damage more easily.
This fight can be very frustrating, but take heart in knowing that you're close to the end of the game. Return to Firelink Shrine and chuck Lothric's head on the last throne to kick things off with the Fire Keeper.
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