Hyperbolic beta integrals are analogues of Euler's beta integral in which the role of Euler's gamma function is taken over by Ruijsenaars' hyperbolic gamma function. They may be viewed as $(q,\widetilde{q})$-bibasic analogues of the beta integral in which the two bases $q$ and $\widetilde{q}$ are interrelated by modular inversion, and they entail $q$-analogues of the beta integral for $|q|=1$. The integrals under consideration are the hyperbolic analogues of the Ramanujan integral, the Askey-Wilson integral and the Nassrallah-Rahman integral. We show that the hyperbolic Nassrallah-Rahman integral is a formal limit case of Spiridonov's elliptic Nassrallah-Rahman integral.

35 pages. Remarks and references to recent new developments are added. To appear in Adv. Math