We consider numerical analysis of Bayesian estimation for uncertain systems governed by partial or ordinary differential equations, with uncertain parameters in high-dimensional parameter spaces, subject to given measured data which is assumed to be subject to additive Gaussian observation noise. Bayesian estimation, conditional to given data, then takes the form of a mathematical expectation of system responses over all possible realizations of the uncertain input, conditional on the measurement data. For uncertainties in an infinite-dimensional function space which admits an unconditional basis, this expectation is written as infinite-dimensional iterated integral w.r. to the prior measure. Regularity of the uncertain input translates into a sparsity result of the Bayesian posterior density. Based on this sparsity, we survey several dimension-adaptive quadrature approaches that allow the approximate evaluation of the Bayesian estimates with convergence rates that depend only on the sparsity, and that are independent of the dimension of the parametric domain. Numerical examples from PDEs with random inputs, shape-uncertainty, and large nonlinear systems of parametric ODEs arising in biological systems engineering illustrate applications of the theory. Work supported by ERC under AdG247277 and by SNF.