  These notes are the output of a decade of research on how the results about
dilations of one-parameter CP-semigroups with the help of product systems, can
be put forward to d-parameter semigroups - and beyond. While exisiting work on
the two- and d-parameter case is based on the approach via the
Arveson-Stinespring correspondence of a CP-map by Muhly and Solel (and limited
to von Neumann algebras), here we explore consequently the approach via
Paschke's GNS-correspondence of a CP-map by Bhat and Skeide. (A comparison is
postponed to Appendix A(iv).)
  The generalizations are multi-fold, the difficulties often enormous. In fact,
our only true if-and-only-if theorem, is the following: A Markov semigroup over
(the opposite of) an Ore monoid admits a full (strict or normal) dilation if
and only if its GNS-subproduct system embeds into a product system. Already
earlier, it has been observed that the GNS- (respectively, the
Arveson-Stinespring) correspondences form a subproduct system, and that the
main difficulty is to embed that into a product system. Here we add, that every
dilation comes along with a superproduct system (a product system if the
dilation is full). The latter may or may not contain the GNS-subproduct system;
it does, if the dilation is strong - but not only.
  Apart from the many positive results pushing forward the theory to large
extent, we provide plenty of counter examples for almost every desirable
statement we could not prove. Still, a small number of open problems remains.
The most prominent: Does there exist a CP-semigroup that admits a dilation, but
no strong dilation? Another one: Does there exist a Markov semigroup that
admits a (necessarily strong) dilation, but no full dilation?
