  We survey determinantal singularities, their deformations, and their
topology. This class of singularities generalizes the well studied case of
complete intersections in several different aspects, but exhibits a plethora of
new phenomena such as for instance non-isolated singularities which are
finitely determined, or smoothings with low connectivity; already the union of
the coordinate axes in $(\mathbb{C}^3,0)$ is determinantal, but not a complete
intersection. We start with the algebraic background and then continue by
discussing the subtle interplay of unfoldings and deformations in this setting,
including a survey of the case of determinantal hypersurfaces, Cohen-Macaulay
codimension $2$ and Gorenstein codimension $3$ singularities, and determinantal
rational surface singularities. We conclude with a discussion of essential
smoothings and provide an appendix listing known classifications of simple
determinantal singularities.
