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We give a survey at an introductory level of old and recent results in thestudy of critical points of solutions of elliptic and parabolic partialdifferential equations. To keep the presentation simple, we mainly considerfour exemplary boundary value problems: the Dirichlet problem for the Laplace'sequation; the torsional creep problem; the case of Dirichlet eigenfunctions forthe Laplace's equation; the initial-boundary value problem for the heatequation. We shall mostly address three issues: the estimation of the localsize of the critical set; the dependence of the number of critical points onthe boundary values and the geometry of the domain; the location of criticalpoints in the domain.

An introduction to the study of critical points of solutions of elliptic and parabolic equations