The theory of the first-order lattice vibration Raman effect is given with particular reference to semiconductors. It is shown that the most important Raman scattering mechanism is always one in which the radiation interacts indirectly with the lattice via the electrons. The electron-lattice interaction is treated by the deformation potential approximation, and the additional long-range electrical interaction is included for the polar semiconductors. For both types of interaction, the proportion of incident photons Raman scattered by the optic phonons into unit solid angle is of order 1 in 10$^6$ or 10$^7$ for the conventional experimental geometry. The symmetry and frequency dependence of the scattering tensor are derived, and expressions are obtained for the width and temperature shift of the Raman lines. For acoustic phonons it is shown that the present theory correctly reproduces the previously established theoretical Raman intensities. Microscopic expressions for the elasto-optical and electro-optical coefficients are derived as by-products of the calculation.