Briefly defining differential subordination in the complex plane as the generalization of a differential inequality on the real line, Miller (mathematics, State U. of New York-Brockport) and Mocanu (complex analysis, Baves-Bolyai U., Cluj-Napoca, Romania) combine the basic concepts with recent results in such fields as differential equations, partial differential equations, meromorphic functions, harmonic functions, integral operators, Banach spaces, and functions of several complex variables. They describe the theory of first-order and second-order differential subordinations with some results from higher orders, simplify many of the proofs, extend the results of univalent function theory, and explain the relationship between differential subordinations and differential equations. The text is double spaced.