I propose novel methods for understanding key aspects that are essentialto the future of Bayesian inference for high- or infinite-dimensionalmodels and data. By combining my expertise on empirical processes andlikelihood theory with my recent work on posterior contraction I shallforemost lay a mathematical foundation for the Bayesian solution touncertainty quantification in high dimensions.Decades of doubt that Bayesian methods can work for high-dimensionalmodels or data have in the last decade been replaced by a belief thatthese methods are actually especially appropriate in thissetting. They are thought to possess greater capacity forincorporating prior knowledge and to be better able to combine datafrom related measurements. My premise is that for high- orinfinite-dimensional models and data this belief is not well founded,and needs to be challenged and shaped by mathematical analysis.My central focus is the accuracy of the posterior distribution asquantification of uncertainty. This is unclear and has hardly beenstudied, notwithstanding that it is at the core of the Bayesianmethod. In fact the scarce available evidence on Bayesian crediblesets in high dimensions (sets of prescribed posterior probability)casts doubt on their ability to capture a given truth. I shall discoverhow this depends strongly on the prior distribution, empirical orhierarchical Bayesian tuning, and posterior marginalizaton, and therewithgenerate guidelines for good practice.I shall study these issues in novel statistical settings (sparsity andlarge scale inference, inverse problems, state space models,hierarchical modelling), and connect to the most recent, excitingdevelopments in general statistics.I work against a background of data-analysis in genetics, genomics,finance, and imaging, and employ stochastic process theory,mathematical analysis and information theory.