This thesis is dedicated to developing a dilation theory for semigroups of
completely positive maps. The first part treats two-parameter semigroups, and
contains also contributions to dilation theory of product system
representations. The second part deals with completely positive semigroups
parameterized by quite general semigroups, where the major technical tool
introduced is subproduct systems and their representations. In the third part
subproduct systems are studied, together with the multivariable operator theory
and operator algebras they give rise to.