  The aim of the lecture is to briefly describe the mathematical background of
scattering theory for two- and three-particle quantum systems. We discuss basic
objects of the theory: wave and scattering operators and the corresponding
scattering matrix and illustrate them on the example of the Schr\"odinger
equation. Our goal is to present time-dependent and stationary approaches and
to describe the underlying mathematical methods. We also give a sketch of
scattering theory for three interacting quantum particles including a difficult
problem of the asymptotic completeness of scattering channels. Along with
traditional results, we discuss new scattering channels arising for long-range
pair interactions.
