This is a joint seminar with the PACM Colloquium.  Please note special location.  Many problems in the physical sciences require the determination of an unknown function from a finite set of indirect measurements. Examples include oceanography, medical imaging, oil recovery, water resource management and weather forecasting. Furthermore there are numerous inverse problems where geometric characteristics, such as interfaces, are key unknown features of the overall inversion. Applications include the determination of layers and faults within subsurface formations, and the detection of unhealthy tissue in medical imaging. We describe a theoretical and computational Bayesian framework relevant to the solution of inverse problems for functions, and for geometric features. We formulate Bayes' theorem on separable Banach spaces, a conceptual approach which leads to a form of probabilistic well-posedness and also to new and efficient MCMC algorithms which exhibit order of magnitude speed-up over standard methodologies. Furthermore the approach can be developed to apply to geometric inverse problems, where the geometry is parameterized finite-dimensionally and, via the level-set method, to infinite-dimensional parameterizations. In the latter case this leads to a well-posedness that is difficult to achieve in classical level-set inversion, but which follows naturally in the probabilistic setting.