Path integral formulation
Feynman developed the path integral formulation of quantum mechanics in 1948, using the following postulates:
1. The probability for any fundamental event is given by the absolute square of a complex amplitude.
2. The amplitude for some event is given by adding together all the histories which include that event.
3. The amplitude a certain history contributes is proportional to exp((i/hbar)I(H)), where I(H) is the action of that history.
Feynman showed that his formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics. The usual use of the path integral is calculate <q1,t1|q0,t0>, a quantity known as the propagator. As such it is very useful in quantum field theory.
Feynman also developed a simple set of rules for performing quantum calculations which are known as Feynman diagrams.
Feynman's postulates are somewhat ambigous in that they do not define what an "event" is or the exact proportionality constant in 3. The proportionality problem can be solved by simply normalizing the path integral by dividing the amplitude by the square root of the total probability for something to happen (resulting in that the total probability given by all the normalized amplitudes will be 1, as we would expect). Generally speaking one can simply define the "events" in an operational sense for any given experiment. In one philosophical interpretation of quantum mechanics, the "sum over histories" interpretation, the path integral is taken to be fundamental and reality is viewed as a single indistinguishable "class" of paths which all share the same events. For this interpretation, it is crucial to understand what exactly an event is. Despite its general unpopularity, the sum over histories method gives identical results to canonical quantum mechanics and also explains the Einstein-Podolsky-Rosen paradox without resorting to nonlocality (see A Sum-over-histories Account of an EPR(B) Experiment {Found. of Phys. Lett.} {4} {303-335} {1991}, http://physics.syr.edu/~sorkin/some.papers/63.eprb.ps). This makes it the only form of the theory which can explain this paradox without breaking locality.