Wave maps are the natural generalization of solutions of the linear wave equation to mappings between (pseudo-)Riemannian manifolds. After several groundbreaking results in the previous decade, wave maps from Minkowski spacetime $\mathbf{R}^{1+n}$ are now well-understood, including the particularly delicate energy-critical case $n=2$. I will give an overview of this area and the main technical challenges, and also discuss some more recent progress in extending these results to curved domains.