  We describe a new method of calculation of generic multi-loop master
integrals based on the numerical solution of systems of difference equations in
one variable. We show algorithms for the construction of the systems using
integration-by-parts identities and methods of solutions by means of expansions
in factorial series and Laplace's transformation. We also describe new
algorithms for the identification of master integrals and the reduction of
generic Feynman integrals to master integrals, and procedures for generating
and solving systems of differential equations in masses and momenta for master
integrals. We apply our method to the calculation of the master integrals of
massive vacuum and self-energy diagrams up to three loops and of massive vertex
and box diagrams up to two loops. Implementation in a computer program of our
approach is described. Important features of the implementation are: the
ability to deal with hundreds of master integrals and the ability to obtain
very high precision results expanded at will in the number of dimensions.
