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

Resolvable Cycle Decompositions of Complete Multigraphs and Complete Equipartite Multigraphs via Layering and Detachment