The basic idea of inflation in cosmology is very simple: It is the assumption that the expansion factor R(t) of a Friedmann-Lemaltre cosmological model grows exponentially during a brief time interval in the very early universe. The phase of exponential growth is followed by a thermalizatlon stage and a subsequent “normal” evolution R(t)∼vt. This “Inflationary expansion“ can help to solve cosmological puzzles inherent in the standard model - such as the large-scale flatness, the horizon structure, the numerical value of the entropy in a comoving volume [for a review see Brandenberger 1985]. To turn this romantic idea of inflation into a quantitative model requires still a lot of work: The simple change in the thermal history of the universe must be derived from a fundamental particle theory. The models proposed so far do not inspire much confidence. In the following a few difficulties of the Higgs field idea, especially the Coleman-Weinberg formalism will be pointed out (section 1). In section 2 some problems connected with the investigation of initially strongly anisotropic or Inhomogeneous cosmological models will be mentioned.