Radiation force is a universal phenomenon in any wave motion, electromagnetic or acoustic. Although acoustic and electromagnetic waves are both characterized by time variation of basic quantities, they are also both capable of exerting a steady force called radiation force. In 1902, Lord Rayleigh published his classic work on the radiation force of sound, introducing the concept of acoustic radiation pressure, and some years later, further fundamental contributions to the radiation force phenomenon were made by L. Brillouin and P. Langevin. Many of the studies discussing radiation force published before 1990 were related to techniques for measuring acoustic power of therapeutic devices; also, radiation force was one of the factors considered in the search for noncavitational, nonthermal mechanisms of ultrasonic bioeffects. A major surge in various biomedical applications of acoustic radiation force started in the 1990s and continues today. Numerous new applications emerged including manipulation of cells in suspension, increasing the sensitivity of biosensors and immunochemical tests, assessing viscoelastic properties of fluids and biological tissues, elasticity imaging, monitoring ablation of lesions during ablation therapy, targeted drug and gene delivery, molecular imaging and acoustical tweezers. We briefly present in this review the major milestones in the history of radiation force and its biomedical applications. In discussing the physical basis of radiation force and its applications, we present basic equations describing the relationship of radiation stress with parameters of acoustical fields and with the induced motion in the biological media. Momentum and force associated with a plane-traveling wave, equations for nonlinear and nonsteady-state acoustic streams, radiation stress tensor for solids and biological tissues and radiation force acting on particles and microbubbles are considered.