  In this paper, we discuss f- and flag-vectors of 4-dimensional convex
polytopes and cellular 3-spheres. We put forward two crucial parameters of
fatness and complexity: Fatness F(P) := (f_1+f_2-20)/(f_0+f_3-10) is large if
there are many more edges and 2-faces than there are vertices and facets, while
complexity C(P) := (f_{03}-20)/(f_0+f_3-10) is large if every facet has many
vertices, and every vertex is in many facets. Recent results suggest that these
parameters might allow one to differentiate between the cones of f- or
flag-vectors of -- connected Eulerian lattices of length 5 (combinatorial
objects), -- strongly regular CW 3-spheres (topological objects), -- convex
4-polytopes (discrete geometric objects), and -- rational convex 4-polytopes
(whose study involves arithmetic aspects). Further progress will depend on the
derivation of tighter f-vector inequalities for convex 4-polytopes. On the
other hand, we will need new construction methods that produce interesting
polytopes which are far from being simplicial or simple -- for example, very
``fat'' or ``complex'' 4-polytopes. In this direction, I will report about
constructions (from joint work with Michael Joswig, David Eppstein and Greg
Kuperberg) that yield -- strongly regular CW 3-spheres of arbitrarily large
fatness, -- convex 4-polytopes of fatness larger than 5.048, and -- rational
convex 4-polytopes of fatness larger than 5-epsilon.
