A new class of copulas referred to as "Sibuya copulas" is introduced and its properties are investigated.The construction of Sibuya copulas is based on an increasing stochastic process whose Laplace-Stieltjes transform enters the copula as a parameter function.Sibuya copulas also allow for idiosyncratic parameter functions and are thus quite flexible to model asymmetric dependencies.If the stochastic process is allowed to have jumps,Sibuya copulas may have a singular component.Depending on the choice of the process,several well-known examples arise,for example,extreme-value copulas,Lévy-frailty copulas,or Marshall-Olkin copulas.Furthermore,as a special homogeneous case,one may obtain any Archimedean copula with Laplace-Stieltjes transform generator.Detailed examples are given.Sibuya copulas arise naturally in terms of a generalization of intensity-based default models where the deterministic term structure of the survival probability is made stochastic.This interpretation allows for a sampling algorithm.It can easily be generalized to allow for hierarchies.