What are some examples of non-Von Neumann architectures?

If I understand correctly modern computers are modeled after the Von Neumann architecture. I have sometimes seen reference to alternatives, but haven't really seen any very good descriptions of how non-Von Neumann architectures would be organised and function.

Does anyone have any examples? What are the advantages/disadvantages of alternative computer organization?

Examples of non von Neumann machines are the dataflow machines and the reduction machines. In both of these cases there is a high degree of parallelism, and instead of variables there are immutable bindings between names and constant values. non von Neumann is usually reserved for machines that represent a radical departure from the von Neumann model, and is therefore not normally applied to multiprocessor or multicomputer architectures, which effectively offer a set of cooperating von Neumann machines.

I believe the most common one would be the Harvard architecture or the Modified Harvard architecture which is used in a lot of ARM based chips. I am sure there are many differences, but here is one that stands out

In a computer using the Harvard architecture, the CPU can both read an instruction and perform a data memory access at the same time, even without a cache.

Cellular automata - this predated neural network, but the ideas are very similar.

http://en.wikipedia.org/wiki/Cellular_automaton

Neural networks can be viewed as a form of cellular automata, or a distinct non-von Neuman architecture of their own.

http://en.wikipedia.org/wiki/Neural_network

Quantum computers are also another example. They differ from other computers, mainly because of the order of time needed for a problme to be solved on them. Usual computers can be modeld using Turing machine whether von Neuman or Harvard. They have their own model called Quantum turing machine.