The Casimir effect is analyzed. This effect consists of a polarization of the vacuum of quantized fields which arises as a result of a change in the spectrum of vacuum oscillations when the quantization volume is bounded or the topology of the space is non-Euclidean. Calculations of the effect for manifolds of various configurations and for fields with various spins are reported. Various definitions of the Casimir vacuum energy in the presence of walls are discussed. The quantum field theory of Casimir forces is generalized to incorporate the dispersion properties of the medium. Applications of the Casimir effect in various branches of physics are reviewed: from the theory of molecular forces to cosmology and elementary particle physics, including the bag model and supersymmetry.