A categorical subset of a space X is a subset which is contractible in
X. The Lusternik-Schnirelmann category of X measures how many open
categorical sets you need to cover X. For a closed smooth manifold X it
provides a lower bound for the number of critical points any smooth
function on X must have. The first part of the talk illustrates some
results on this invariant. In the second part we explain how this
concept generalizes to foliated manifolds, what its meaning is, and we
survey some recent results.