We discuss the physics of diffusive compression acceleration of charged particles. This mechanism is similar to diffusive shock acceleration, which is well known, except that it applies to a gradual compression of the plasma, rather than a shock. It also applies to fluctuations in velocity, such as turbulence. We present the results from theoretical and numerical calculations for a variety of applications of this mechanism. These include: acceleration of particles by corotating solar-wind compressions, acceleration in a sinusoidally varying fluid flow, the formation of high-energy tails in pickup-ion distributions, and the acceleration of Galactic cosmic rays by turbulent flows in the interstellar medium. We also show that this mechanism naturally favors acceleration of particles with larger gyroradii. This may help explain recent observations of some solar-energetic particle observations showing abundance enhancements that increase with particle gyroradius.