Expert is a very challenging mode, if you're new to it. There WILL be restarts so please, do not give up and try to complete through it. It will frustrate you, but you cannot give up. Mistakes will be made, which is good on single-player or with friends. Public matches, people won't take your mistakes lightly, and they will votekick you right out of the game, so please, PLEASE, I encourage you play on single-player or with friends. I may sound like a broken record, but I don't want the new players to feel discouraged and to never touch this game again just because the community comes off as unpleasant.

Common Infected
The common infecteds do 20 damage when you're facing them, and 10 when they're behind you.
That's 5 swipes in the front and 10 in the back, which is BRUTAL. You must shove (pressing M2 or also known as your right-mouse button) to get them off you, and always check your back.

Also, turn on your flashlights in dark areas and check corners. You could be at 10 health and a common infected will take just a swipe at you, making you get incapped.

Tank
A tank on expert (and as well advanced) has 8 thousand health, making it harder to kill. So grab a molly and light it up!

Note: a tank can incap you in ONE HIT! Using a rock, punch, whatever! So don't get trapped in a corner, juke that tank (Juking means to dodge a tank's attacks by moving and jumping.), and use your medical items to kite easier.

Witch
If you startle her, you're dead. Literally dead. One hit will not incap you, it will bring you straight to death.
However, she'll still have the same amount of HP like in the other modes.

Now, on to the expert tips. Like I said before, this can be applied to many other gamemodes.

-When a horde appears, quickly get to a corner or a small room.
Do not try to progress. Once a horde spawns, look around quickly. Corner, bathroom, office, etc. Make sure that you are safe and wait until the horde dies out.

If your teammate(s) get incapped during the horde, do NOT save them. Wait until the horde dies out. If you go out and try to save them, YOU'LL be the one to get incapped as well. If your teammate(s) have a problem with that, they're most likely inexperienced.

-Conserve medical items
Medkits: Gives you 80% HP upon using. Use it only at these situations
-Black & white (BW)
-A tank spawns (25 HP suggested)
-Right before an event (39 HP suggested, and no pills or shots are found.)
-Once you spot an extra medkit (lowest HP player heals up and takes the medkit. Don't be selfish and take it for yourself at green HP.)

Medkits are critical in surviving the whole campaign. If you just waste them, you're going to have a very tough time getting through it. So, conserve your medkits to insure your great chances of survival in the campaign.

Pills: Gives you 50 temp HP upon using. Use it only at these situations
-Red health
-A tank spawns
-When a horde appears
-When you spot an extra bottle of pills

Pills can keep the team moving without having to waste medkits.
(Please note that pills do not cure your black and white condition. You must use a medkit. Also, don't use pills right outside the saferoom)

Adrenaline Shot: Gives you 25 temp HP and a speed boost upon using. Use it only at these situations
-40 and below HP
-A tank spawns
-When a horde appears
-When you spot an extra bottle of pills or an extra shot
-Right before an event

Not much to say about the adrenaline shot. You'll still survive without em, but it'll be difficult, especially the run on "Dark Carnival", or the bridge run on "The Parish".

Always crouch while shooting
If you crouch, you'll get more accurate than standing. Also, your teammates can shoot over you so you won't take friendly damage! Very useful during a horde.

Pick your best weapon
If your favorite weapon is the M16, use it. If your favorite weapon is the Autoshotgun, use it. Only use the guns that you are familar and comfortable with. This can help you make it through the expert campaign in a breeze.

On running events, equip the shotgun or sniper
*As suggested in the comments. Just modified*
While on the move, the shotgun and sniper are most effective to take out crowds while running off. Just please, don't run off too fast or do any FF.

If you hear a special infected, get your squad close and shoot it dead together
*As suggested in the comments.
It IS possible to take out 3 specials alone, but it's more safer and quicker to do it together as a team. So make sure you alert your team that a special infected is close-by and take it out together.

Conserve grenades
Bile: These are the grenades that can literally make or break a game, depending on which campaign you're playing on. Seriously, you cannot use them as if it's easy/normal. You HAVE to grab it and save it when you think it's best to be used.
An example is at the 1st chapter of Dark Carnival. Once you pass that chapter, no biles will appear. So make sure you find a bile and keep it until the coaster or the gate run.
(Note: Please don't use them on tanks. Once that horde stops attacking the tank, guess what? It's coming after you next.)

Pipebomb: These are only to be used when someone is biled, a horde comes, or during an event. Yes, that's basically it.
Don't toss it at the tank.
Just.
Don't.

Yes, the pipebomb can and WILL stumble a tank when thrown at him. However, it's a waste to do so, as a pipebomb can be used later on to kill out a huge horde down the line. If I were you, I would just instead pick up a propane tank and toss it at the tank and stumble him like that, as it does far more damage to him anyway.

Molotov: On expert, these things are DEADLY. One missed toss can lead to death if teammates are in the flames. Only use these things when a tank appears or a huge horde appears.

Ammo Packs
Frag rounds - Only use these when a horde comes. Don't shoot at a zombie when they come too close to you though, you'll get damaged from the blast.
Also useful aganist witches and special infected.

Fire ammo - Use these on a horde, or a tank.
Please note: the fire lit on a tank or a special will NOT be permanent. It will go away after 5 seconds.

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List of updates made for this guide:
Update #1, June 3rd 2016, 8:47 PM EST:
-Fixed information regarding health kits. They heal 80 PERCENT of your health, not 80 HP.
- Added some new information in the grenade and medical supply sections.
- Simple fixes on grammar.
- Deleted unnecessary information for the expert mode.