1: A Lizard in a Woman's Skin
2: Don't Torture a Duckling
3: The Psychic
4: Four of The Apocalypse
5: The Beyond
6: Zombi 2

Seven Notes in Black
Fou5b4r of the Apocalypse
Beatrice Cenci
New York Ripper
The Beyond

1. The Beyond
2. Zombi 2
3. A Lizard in a Woman's Skin
4. City of the Living Dead
5. The New York Ripper
   Have you seen the little piggies crawling in the dirt?

No special order:
"The devil's honey"
"Voices from beyond"
"The beyond"
"Don't torture a duckling"
"Four of the apocalypse"
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http://darioargentofr.blogspot.com/

1. The Beyond
2. Don't Torture a Duckling
3. Zombi 2
4. The Gates of Hell
5. Four of the Apocalypse
   I am the Duke of IMDb bio writers! I am A#1!

1. City of the Living Dead
2. One on Top of the Other
3. Four of the Apocalypse
4. Zombie Flesh Eaters
5. A Lizard in a Woman's Skin

1. The Beyond (best Italian horror)
2. City Of The Living Dead (PURE horror, my favourite gore scenes)
3. One On Top Of The Other (Awesome thriller, great music, I love those split screen scenes)
4. Seven Notes In Black (Beautiful music, great scenes)
5. Zombi 2 (Campy classic)

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Your answer implies that you have not seen more than 1 or 2 of his films.

What do people rate as his best five?
The question falsely implies he made five films worth sitting through.

1 Lizard In A Woman's Skin
2.Don't Toture A Duckling
3.Seven Notes In Black aka The Psychic
4. One On Top of The Other aka Perversion Story
5. Beatrice Cenci

In no order at all
1.City of the Living Dead
2.The Beyond
3.Don't Torture a Duckling
4.The House by the Cemetery
5.The New York Ripper

I have seen 25 now and this how I would rank them (Some rating changed due to rewatches). I have another 7 on order so I will update after I see those.
1 City of the Living Dead
2 Beatrice Cenci
3 Conquest
4 A Lizard in a Woman's Skin
5 Four of the Apocalypse
6 Contraband
7 House by the Cemetery
8 Seven Notes in Black
9 Zombi 2
10 The Beyond
11 The New York Ripper
12 Don't Torture a Duckling
13 Dangerous Obsession (Devil's Honey)
14 Touch of Death
15 Young Dracula
16 Massacre Time
17 Manhattan Baby
18 House of Clocks
19 Sodoma's Ghost
20 The New Gladiators
21 Perversion Story
22 Demonia
23 The Black Cat
24 A Bullet for Sandoval (does this one count as a Fulci film?)
25 Door To Silence

I'll post this now having only seen something like 12 or 13 of his films, I bet that my opinions will change after I see more.

1. Beatrice Cenci
2. A Lizard in a Woman's Skin
3. City of the Living Dead
4. Conquest
5. Seven Notes in Black

Of the ones I've seen so far, I rate them like this-
Zombi 2
New York Ripper
Murder Rock
Lizard in a Woman's Skin
House By the Cemetary
City of the Living Dead
"Bad luck isn't brought by broken mirrors, but by broken minds" - Dario Argento's SUSPIRIA.

The trouble with this simple question is that Fulci went through such a stylistic evolution that most of his films are appreciated in a way difficult to compare them with one another.How about this
His best horror. (best Italian horror ever) the Beyond, but you shouldn't miss Zombi 2
best Giallo: Don't torture a duckling
Best "relationship" film: Devil's Honey
Best Crime: Contraband
Best Western: Four of the apocalypse
Comedy: Senator Likes Women
Fulci was at his best when working with Segio Salvati anf Fabio Frizzi

in no order
the beyond(a masterpiece i dont class this as a zombie movie)
house by the cemetery(freudstein is so creepy and owns!)
zombie flesh eaters (amazing set pieces and my fave zombie film ever)
city of the living dead(so atmospheric very dark and rather scary)
newyork ripper (by far one the most violent films if not the most violent hes made but still very suspensful in some scenes)
lucio fulci GOD FARTHER OF GORE R.I.P