This accessible and self-contained treatment provides even readers previously unacquainted with parabolic and elliptic equations with sufficient background to understand research literature. Author Avner Friedman -- Director of the Mathematical Biosciences Institute at The Ohio State University -- offers a systematic and thorough approach that begins with the main facts of the general theory of second order linear parabolic equations. Subsequent chapters explore asymptotic behavior of solutions, semi-linear equations and free boundary problems, and the extension of results concerning fundamental solutions and the Cauchy problem to systems of parabolic equations. The final chapter concerns questions of existence and uniqueness for the first boundary value problem and the differentiability of solutions, in terms of both elliptic and parabolic equations. The text concludes with an appendix on nonlinear equations and bibliographies of related works.