We develop a theory of bornological quantum hypergroups, aiming to extend the theory of algebraic quantum hypergroups in the sense of Delvaux and Van Daele to the framework of bornological vector spaces. It is very similar to the theory of bornological quantum groups established by Voigt, except that the coproduct is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterized in terms of these integrals. We study the Fourier transform and develop Pontryagin duality theory for a bornological quantum hypergroup. As an application, we prove a formula relating the fourth power of the antipode with the modular functions of a bornological quantum hypergroup and its dual.