  The Steklov eigenvalue problem, first introduced over 125 years ago, has seen
a surge of interest in the past few decades. This article is a tour of some of
the recent developments linking the Steklov eigenvalues and eigenfunctions of
compact Riemannian manifolds to the geometry of the manifolds. Topics include
isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the
case of surfaces and then in higher dimensions), stability and instability of
eigenvalues under deformations of the Riemannian metric, optimisation of
eigenvalues and connections to free boundary minimal surfaces in balls, inverse
problems and isospectrality, discretisation, and the geometry of
eigenfunctions. We begin with background material and motivating examples for
readers that are new to the subject. Throughout the tour, we frequently compare
and contrast the behavior of the Steklov spectrum with that of the Laplace
spectrum. We include many open problems in this rapidly expanding area.
