  The foundations of matrix geometry are discussed, which provides the basis
for recent progress on the effective geometry and gravity in Yang-Mills matrix
models. Basic examples lead to a notion of embedded noncommutative spaces
(branes) with emergent Riemannian geometry. This class of configurations turns
out to be preserved under small deformations, and is therefore appropriate for
matrix models. The relation with spectral geometry is discussed. A possible
realization of sufficiently generic 4-dimensional geometries as noncommutative
branes in D=10 matrix models is sketched.
