  We give a survey at an introductory level of old and recent results in the
study of critical points of solutions of elliptic and parabolic partial
differential equations. To keep the presentation simple, we mainly consider
four exemplary boundary value problems: the Dirichlet problem for the Laplace's
equation; the torsional creep problem; the case of Dirichlet eigenfunctions for
the Laplace's equation; the initial-boundary value problem for the heat
equation. We shall mostly address three issues: the estimation of the local
size of the critical set; the dependence of the number of critical points on
the boundary values and the geometry of the domain; the location of critical
points in the domain.
