  We construct new resolvable decompositions of complete multigraphs and
complete equipartite multigraphs into cycles of variable lengths (and a perfect
matching if the vertex degrees are odd). We develop two techniques: {\em
layering}, which allows us to obtain 2-factorizations of complete multigraphs
from existing 2-factorizations of complete graphs, and {\em detachment}, which
allows us to construct resolvable cycle decompositions of complete equipartite
multigraphs from existing resolvable cycle decompositions of complete
multigraphs. These techniques are applied to obtain new 2-factorizations of a
specified type for both complete multigraphs and complete equipartite
multigraphs, with the emphasis on new solutions to the Oberwolfach Problem and
the Hamilton-Waterloo Problem. In addition, we show existence of some
$\alpha$-resolvable cycle decompositions.
