Yes, but there is no single set of zombie rules, and no one is obliged to follow any rules whatsoever.
The most popular "zombie rules" are the so-called "Romero rules", named for the director of the first modern zombie film, Night of the Living Dead, George Romero. However, the rules were not invented by Romero, and his later films break most of the rules named after him. These rules are basically as follows:
Zombies are dead
Zombies are slow
Zombies are stupid
Zombies eat living flesh, or recently killed flesh, especially human flesh
Zombies can only be killed by the destruction of the brain
Being bitten by a zombie turns you into a zombie
These very vague rules were later expanded and explained more thoroughly by subsequent writers and directors, most notably Max Brooks, who wrote The Zombie Survival Guide and World War Z (the book, not the movie). The expanded rules have come to include such items as:
And so on. Because Brooks put so much effort and attention to detail into his work, and because Romero's subsequent films have been so horrible, I actually prefer to use the term "Brooks Rules" rather than "Romero Rules". As it happens, The Walking Dead follows Romero/Brooks Rules, but the World War Z movie follows "Fast Zombie Rules" (see below for more information).
There are other types of zombies, but the rules, such as they are, are much looser.
One alternative set of zombie rules is known as "Fast Zombie Rules":
Zombies are still stupid
Zombies are still dead
Zombies are still cannibals
Zombies are somewhat uncoordinated, but are quite fast
Zombies still have to be killed by destroying the brain
Bites still turn you into a zombie
In fact, zombies can be/do anything you want them to. So we get fast zombies, talking zombies, smart(ish) zombies, etc.
I don't know if you could consider this a formal set of zombie rules, but there are what are called "Return of the Living Dead Zombies", for the schlocky comedy horror movie of the same name. The general idea behind this type of zombie is as follows:
Zombies can talk, but not very well (most of the time, they basically just moan "Braaaaaaaains!" constantly, but sometimes they are more articulate and even clever and tricky)
Zombies eat brains, not just flesh
Zombies retain some elements of their former personalities
Zombies can do anything necessary to obtain a cheap laugh from the movie audience
Otherwise, the Romero/Brooks Rules still apply
As far as 28 Days/Weeks Later is concerned, no, they aren't zombies. These are living people infected with the so-called "Rage Virus". They die very quickly, because they don't eat or drink anything, and they can be killed the same way regular people can be killed. They are a bit like zombies, in that they are stupid and aggressive and infect others through bodily fluid transfers, but strictly speaking, they do not meet the requirements for zombieness. I don't have time to find it now, but I believe the director of the 28 ______ Later movies specifically said that the infected people I'm his films are not zombies, and were never intended to be zombies.
If you are interested in learning more about zombie rules, the best place to begin is with Max Brooks' books. * World War Z* is far better than the movie, and is more readable than his other book, The Zombie Survival Guide, which is exactly what it sounds like - it reads like any other survival guide, but with zombies. It can be a bit tedious at times, and isn't much fun to read, unless you are a true zombie fanatic.