Atomic polarization phenomena impinge upon a number of areas and processes in physics. The dielectric constant and refractive index of any gas are examples of macroscopic properties that are largely determined by the dipole polarizability. When it comes to microscopic phenomena, the existence of alkaline-earth anions and the recently discovered ability of positrons to bind to many atoms are predominantly due to the polarization interaction. An imperfect knowledge of atomic polarizabilities is presently looming as the largest source of uncertainty in the new generation of optical frequency standards. Accurate polarizabilities for the group I and II atoms and ions of the periodic table have recently become available by a variety of techniques. These include refined many-body perturbation theory and coupled-cluster calculations sometimes combined with precise experimental data for selected transitions, microwave spectroscopy of Rydberg atoms and ions, refractive index measurements in microwave cavities, ab initio calculations of atomic structures using explicitly correlated wavefunctions, interferometry with atom beams and velocity changes of laser cooled atoms induced by an electric field. This review examines existing theoretical methods of determining atomic and ionic polarizabilities, and discusses their relevance to various applications with particular emphasis on cold-atom physics and the metrology of atomic frequency standards.