Interference with atomic and molecular matter waves is a rich branch of atomic physics and quantum optics. It started with atom diffraction from crystal surfaces and the separated oscillatory fields technique used in atomic clocks. Atom interferometry is now reaching maturity as a powerful art with many applications in modern science. In this review we first describe the basic tools for coherent atom optics including diffraction by nanostructures and laser light, three-grating interferometers, and double wells on AtomChips. Then we review scientific advances in a broad range of fields that have resulted from the application of atom interferometers. These are grouped in three categories: (1) fundamental quantum science, (2) precision metrology and (3) atomic and molecular physics. Although some experiments with Bose Einstein condensates are included, the focus of the review is on linear matter wave optics, i.e. phenomena where each single atom interferes with itself.