This book gives a systematic treatment of the basic theory of analytic semigroups and abstract parabolic equations in general Banach spaces, and of how such a theory may be used in parabolic PDE's. It takes into account the developments of the theory during the last fifteen years, and it is focused on classical solutions, with continuous or Holder continuous derivatives. On one hand, working in spaces of continuous functions rather than in Lebesgue spaces seems to be appropriate in view of the number of parabolic problems arising in applied mathematics, where continuity has physical meaning; on the other hand it allows one to consider any type of nonlinearities (even of nonlocal type), even involving the highest order derivatives of the solution, avoiding the limitations on the growth of the nonlinear terms required by the LP approach. Moreover, the continuous space theory is, at present, sufficiently well established. For the Hilbert space approach we refer to J. L. LIONS - E. MAGENES [128], M. S. AGRANOVICH - M. l. VISHIK [14], and for the LP approach to V. A. SOLONNIKOV [184], P. GRISVARD [94], G. DI BLASIO [72], G. DORE - A. VENNI [76] and the subsequent papers [90], [169], [170]. Many books about abstract evolution equations and semigroups contain some chapters on analytic semigroups. See, e. g. , E. HILLE - R. S. PHILLIPS [100]' S. G. KREIN [114], K. YOSIDA [213], A. PAZY [166], H. TANABE [193], PH.