Geometric analysis is concerned with the optimization of geometrically defined variational integrals and their applications to the geometry of Riemannian manifolds. Since a Riemannian manifold generically has non-zero curvature, these problems are non-linear. And since they are defined on manifolds of dimensions usually larger than one, the optima have to satisfy partial differential equations.

I will discuss some of the most important such problems that drove the research in the field, harmonic maps and their generalizations, and minimal surfaces or submanifolds.

In dimension two, the former is related to conformal analysis, making the theory particularly rich. And for the latter, we have the Bernstein, Plateau and Dirichlet problems.

I shall present basic techniques and discuss recent research in the field.