The different facets of the R-matrix method are presented pedagogically in
a general framework. Two variants have been developed over the years: (i) The
"calculable" R-matrix method is a calculational tool to derive scattering
properties from the Schr\"odinger equation in a large variety of physical
problems. It was developed rather independently in atomic and nuclear physics
with too little mutual influence. (ii) The "phenomenological" R-matrix
method is a technique to parametrize various types of cross sections. It was
mainly (or uniquely) used in nuclear physics. Both directions are explained by
starting from the simple problem of scattering by a potential. They are
illustrated by simple examples in nuclear and atomic physics. In addition to
elastic scattering, the R-matrix formalism is applied to transfer and
radiative-capture reactions. We also present more recent and more ambitious
applications of the theory in nuclear physics.Comment: 93 pages, 26 figures. Rep. Prog. Phys., in pres