Collisional kinetics is a multiscale phenomenon due to the disparity between the continuum (fluid) and the collisional (particle) length scales. This paper describes a class of simulation methods for gases and plasmas, and acceleration techniques for improving their speed and accuracy. Starting from the Landau-Fokker-Planck equation for plasmas, the focus will be on a binary collision model that is solved using a Direct Simulation Monte Carlo (DSMC) method. Acceleration of this method is achieved by coupling the particle method to a continuum fluid description. The velocity distribution function f is represented as a combination of a Maxwellian M (the thermal component) and a set of discrete particles fp (the kinetic component). For systems that are close to (local) equilibrium, this reduces the number N of simulated particles that are required to represent f for a given level of accuracy. We present two methods for exploiting this representation. In the first method, equilibration of particles in fp, as well as disequilibration of particles from M, due to the collision process, is represented by a thermalization/dethermalization step that employs an entropy criterion. Efficiency of the representation is greatly increased by inclusion of particles with negative weights. This significantly complicates the simulation, but the second method is a tractable approach for negatively weighted particles. The accelerated simulation method is compared with standard PIC-DSMC method for both spatially homogeneous problems such as a bump-on-tail and inhomogeneous problems such as nonlinear Landau damping.