This just in; The zombie apocalypse is already happening, it just hasn't spread to humans yet.
Ok, that may be a little over the top, but stay with me here. There is at least one example in at least one species which would qualify the above statement.
Cordyceps is a fungus family including at least 600 species many of which exhibit a very bizarre method of reproducing. They "take over" the bodies of insects and sometimes even modify the behavior of their hosts in very specific ways.
Ophiocordyceps unilateralis was once considered a member of the Cordyceps family but DNA testing says it's not, however, it is otherwise very similar and like many Cordyceps species it has a very specific host insect and a very specific strategy. It seems to be able to completely control its host arthropod and force it to "do its bidding". Researchers used to believe that this was accomplished by the fungus invading the poor insect's brain. After detailed observation this has proved false. No traces of fungal growth are found in the ant's brain at death. It's now believed that the fungus produces bioactive chemicals and controls the insect's muscles directly.
At first the infected insect, in this case a carpenter ant (Camponotus), appears normal as the fungus is slowly growing inside it. Then, after a few days, it begins to exhibit abnormal behavior.
The infected ant becomes a loner, wandering off of the pheromone marked trail, on its own quest (or somebody's!). Eventually it finds a spot that is perfect, not for the ant's (or colony's) goals, but for the needs of a fungus for optimum spore dispersal. It then finds a large leaf and sinks its mandibles deep into the central vein of the leaf. This is its last living act.
The conditions that the ant/fungus zombie selects are very consistent and very detailed. The leaf selected must be on the north side of the plant, at a height of about 25 cm. (just under 10 inches) from the ground. Temperature needs to be between 20° C and 30° C (68° to 86° F) and the relative humidity about 94 to 95%. The zombie ant then executes its "death grip", dies, and the fungus fruiting body sprouts from the base of its head and fulfills its own mission to release spores. In one instance, Tradd Cotter, an amazing mycologist and researcher who has referred to himself as "a fun guy who studies fungi", noted that the hapless insect he was observing parked its co-opted body directly above a line of ants from its own former colony.
As a side note, a related fungus, Ophiocordyceps sinensis, is well known as both a performance enhancing supplement used by athletes, and as the most expensive fungus in the world. This one uses a moth larva as its host.
I will include a video by National Geographic of the behavior described here for your viewing pleasure. Parental guidance strongly suggested. Also appended are some links for more information.
If there ever is a "zombie apocalypse" attacking humans, I vote for some variant of the bizarre Cordyceps or Ophiocordyceps family of fungi as the "most likely to succeed".
Full Disclosure: After noding this I discovered a similar write-up had been posted by Everyone2. Although there is some overlap, the approaches are different and Everyone2 gives full reproduction rights. Their write-up is pipe linked above.