  These notes provide an introduction to the noncommutative matrix geometry
which arises within matrix models of Yang-Mills type. Starting from basic
examples of compact fuzzy spaces, a general notion of embedded noncommutative
spaces (branes) is formulated, and their effective Riemannian geometry is
elaborated. This class of configurations is preserved under small deformations,
and is therefore appropriate for matrix models. A realization of generic
4-dimensional geometries is sketched, and the relation with spectral geometry
and with noncommutative gauge theory is explained. In a second part, dynamical
aspects of these matrix geometries are discussed. The one-loop effective action
for the maximally supersymmetric IKKT or IIB matrix model is discussed, which
is well-behaved on 4-dimensional branes.
