  Optics and interferometry with matter waves is the art of coherently
manipulating the translational motion of particles like neutrons, atoms and
molecules. Coherent atom optics is an extension of techniques that were
developed for manipulating \emph{internal} quantum states. Applying these ideas
to translational motion required the development of techniques to localize
atoms and transfer population coherently between distant localities. In this
view position and momentum are (continuouse) quantum mechanical degree of
freedom analogous to discrete internal quantum states. In our contribution we
start with an introduction into matter-wave optics in section 1, discuss
coherent atom optics and atom interferometry techniques for molecular beams in
section 2 and for trapped atoms in section 3. In section 4 we then describe
tools and experiments that allow us to probe the evolution of quantum states of
many-body systems by atom interference.
