  This paper outlines a covariant theory of operators defined on groups and
homogeneous spaces. A systematic use of groups and their representations allows
to obtain results of algebraic and analytical nature. The consideration is
systematically illustrated by a representative collection of examples.
  Keywords: Lie groups and algebras, convolution operator, relative
convolution, representation theory, pseudo-differential operators, PDO,
singular integral operator, SIO, the Heisenberg group, SL(2,R), induced
representation, Fock-Segal-Bargmann (FSB) representation, Shrodinger
representation, covariant transform, contravariant transform, wavelet
transform, square integrable representations, reproducing kernel,
Fourier-Wigner transform, deformation quantization, Bergman space, Berezin's
symbol, Wick symbol, Schwartz kernel, Toeplitz operator, operators of local
type, Simonenko's localisation, reproducing kernel thesis, Dynin group, twisted
convolution, star product, Groenewold-Moyal, composition operator.
