We show that Bose-Fermi mixtures constitute promising systems to realise a range of superfluid phases, some with non-trivial topological properties. The fermions attract each other via an induced interaction mediated by the bosons, and the resulting pairing is analysed with retardation effects fully taken into account. For a 3D system, this leads to superfluidity with a critical temperature, which can be high depending on the mass ratio of the bosons and the fermions, and the boson-fermion interaction strength. For a mixed dimensional system, with the fermions living in 2D planes immersed in a 3D Bose-Einstein condensate, we show that the fermions can realise a px+ipy topological superfluid. Since the range of the induced interaction can be tuned experimentally, the critical temperature can approach the maximum value allowed by general Kosterlitz-Thouless theory, without the boson-fermion interaction being particularly strong.