  In these lectures we review two procedures for constructing finite difference
numerical schemes that preserve symmetries of differential equations. The first
approach is based on Lie's infinitesimal symmetry generators, while the second
method uses the novel theory of equivariant moving frames. The advantages of
both techniques are discussed and illustrated with the Schwarzian differential
equation, the Korteweg-de Vries equation and Burgers' equation. Numerical
simulations are presented and innovative techniques for obtaining better
invariant numerical schemes are introduced. New research directions and open
problems are indicated at the end of these notes.
