  These lecture notes treat some current aspects of two closely interrelated
topics from the theory of convex polytopes: the shapes of f-vectors, and
extremal constructions.
  The first lecture treats 3-dimensional polytopes; it includes a complete
proof of the Koebe--Andreev--Thurston theorem, using the variational principle
by Bobenko & Springborn (2004). In Lecture 2 we look at f-vector shapes of very
high-dimensional polytopes. The third lecture explains a surprisingly simple
construction for 2-simple 2-simplicial 4-polytopes, which have symmetric
f-vectors. Lecture 4 sketches the geometry of the cone of f-vectors for
4-polytopes, and thus identifies the existence/construction of 4-polytopes of
high ``fatness'' as a key problem. In this direction, the last lecture presents
a very recent construction of ``projected products of polygons,'' whose fatness
reaches 9-\eps.
