Provided are a curable composition comparable to a curing rate by conventional sulfur while maintaining performance required as a cured product even when a crosslinking agent is replaced with a peroxide, and a sealant containing a cured product of the curable composition. A curable composition containing a conjugated diene-based polymer (A) and a crosslinking agent (B), in which the conjugated diene-based polymer (A) has 100 moles or more of double bonds in a side chain per mole of the polymer.