In this chapter we consider models for collisions between smooth (i.e. frictionless) spherical particles with identical (monodisperse) or different (polydisperse) densities and diameters. For simplicity, we consider only collisions during which the particle mass and diameter are conserved, and exclude other processes that might change these properties (e.g. surface condensation or aggregation (Fox et al., 2008)). Likewise, assuming smooth spherical particles means that the particle angular momentum does not change during a collision, and hence only the particle velocity need be accounted for in the kinetic equation. We limit the discussion to hard-sphere collisions, which implies that the particle velocities after a collision can be written as explicit functions of the particle velocities before the collision, but also consider inelastic collisions with a constant coe.cient of restitution and finite-size particle effects. More details on hard-sphere collisions can be found in the books by Cercignani (Cercignani, 1975, 1988, 2000; Cercignani et al., 1994).We will also briefly discuss simpler collision models that are often used to approximate the hard-sphere collision model in the dilute limit. These include the Maxwell model (Maxwell, 1879) and two linearized collision models (i.e. BGK (Bhatnagar et al., 1954) and ES-BGK (Holway, 1966)) for monodisperse particles, and an extension of the ES-BGK model to polydisperse particles. A discussion of kinetic models for collisions that are not of hard-sphere type can be found in Struchtrup (2005).