In this thesis, we shall give a survey of the initial and initial-boundary value problems for nonlinear parabolic differential equations. For the initial value problem we shall consider various extentions of an old result by H. Fujita for the initial value problem of the nonlinear heat equation
 

u_t = ? u+u^p x ? R^N, p>1

with nonnegative initial values. The main results for initial-boundary value problems, mentioned are due to C. Bandle and H. A. Levine. We will discuss the problems in different geometries.

Keywords: Nonlinear heat equation, Blow up, Critical exponents, Global existence, Global nonexistence