Epoxy resins are typically cured by the use of additives, often called hardeners.  Polyamines are often used.  The amine groups ring-open the epoxide rings.
In rubber, the curing is also induced by the addition of a crosslinker. The resulting process is called sulfur vulcanization. Sulfur breaks down to form polysulfide cross-links (bridges) between sections of the polymer chains. The degree of crosslinking determines the rigidity and durability, as well as other properties of the material.
Paints and varnishes commonly contain oil drying agents, usually metallic soaps that catalyze cross-linking of the unsaturated drying oils that largely comprise them. When paint is described as "drying" it is in fact hardening by crosslinking. Oxygen atoms serve as the crosslinks, analogous to the role played by sulfur in the vulcanization of rubber.