Particle methods are a widely used and well-established class of algorithms for computer simulation in diverse applications, from plasma physics to granular flows. Particle methods can simulate discrete and continuous models stochastically or deterministically, rendering them uniquely versatile. In simulations of discrete models, particles represent discrete entities of the model. In simulations of continuous models, particles represent mathematical collocation or Lagrangian tracer points that discretize continuous fields. Despite their versatility and widespread use, no formal mathematical definition of particle methods was available to unify the different semantics. Here, we present a formal definition of particle methods across discrete and continuous applications, providing a rigorous way of formulating concepts and expressing algorithmic commonalities across applications. This formal definition unifies all particle methods and enables the design of novel particle methods for non-canonical problems. We illustrate the definition by formulating various classic algorithms in this formal framework, from smoothed particle hydrodynamics over molecular dynamics to Gaussian elimination. The definition also enables theoretical analysis of the parallelizability of particle methods on shared- and distributed-memory computers. Finally, we show how a formal mathematical definition of particle methods enables the design and implementation of principled software frameworks for particle methods. Therefore, the presented formal definition of particle methods across discrete and continuous applications opens up new avenues of research for computer simulation algorithms and the corresponding software systems.