Processes with independent, well-behaved increments.
Specific examples of interest include Gamma processes, Brownian Motions, continuous time branching processes etc.
Let's start with George Lowther:
Continuous-time stochastic processes with stationary independent increments are known as Lévy processes. […]it was seen that processes with independent increments are described by three terms — the covariance structure of the Brownian motion component, a drift term, and a measure describing the rate at which jumps occur. Being a special case of independent increments processes, the situation with Lévy processes is similar.[…]
A d-dimensional Lévy process X is a stochastic process taking values in such that
independent increments: is independent of for any
stationary increments: has the same distribution as for any
continuity in probability: in probability as
See change of time