Uniform and tangential harmonic approximation

We give a survey of Runge-type harmonic approximation theorems and the techniques used to prove them. The emphasis is on generalizations of Walsh's classical theorem concerning uniform harmonic approximation on compact sets to the case where approximation takes place on certain non-compact sets: both uniform and tangential approximation are treated. We also give some applications of the theory to the construction of harmonic functions exhibiting various kinds of unexpected behaviour. The course is partly intended to provide preparatory material for S. J. Gardiner's course "Harmonic approximation and applications", published in this volume.