The general formalism of direct nuclear reactions at intermediate energies is developed on the assumption that the reaction proceeds via a knock-out mechanism. Taking the example of an (α, N) reaction, where N stands for a spin ½ particle whose structure can be neglected, calculations are carried out in the distorted wave Born approximation. The effects of the spin-dependent distortions in the initial and final states appear in radial wave functions and phase shifts. A finite-range, spin-independent direct interaction between α and N is used. Explicit formulas are obtained for the differential cross section, the polarization and the angular correlation between the emitted particles and γ-rays from subsequent decay of the final nucleus. Several special cases are treated, such as S-state to S-state transitions, and the simplifications which occur for a zero-range direct interaction are considered. The applicability of the present theory to various type of nuclear reactions is discussed and its possible utility in studying the nuclear reaction mechanism, the cluster structure of nuclei, and the nature of the nuclear surface is pointed out.