We consider the statistical inverse problem of estimating a background flow field (e.g., of air or water) from the partial and noisy observation of a passive scalar (e.g., the concentration of a pollutant). Here the unknown is a vector field that is specified by large or infinite number of degrees of freedom. We show that the inverse problem is ill-posed, i.e., there may be many or no background flows that match a given set of observations. We therefore adopt a Bayesian approach, incorporating prior knowledge of background flows and models of the observation error to develop probabilistic estimates of the fluid flow. In doing so, we leverage frameworks developed in recent years for infinite-dimensional Bayesian inference. We provide conditions under which the inference is consistent, i.e., the posterior measure converges to a Dirac measure on the true background flow as the number of observations of the solute concentration grows large. We also define several computationally-efficient algorithms adapted to the problem. One is an adjoint method for computation of the gradient of the log likelihood, a key ingredient in many numerical methods. A second is a particle method that allows direct computation of point observations of the solute concentration, leveraging the structure of the inverse problem to avoid approximation of the full infinite-dimensional scalar field. Finally, we identify two interesting example problems with very different posterior structures, which we use to conduct a large-scale benchmark of the convergence of several Markov Chain Monte Carlo methods that have been developed in recent years for infinite-dimensional settings. / Ph. D. / We consider the problem of estimating a fluid flow (e.g., of air or water) from partial and noisy observations of the concentration of a solute (e.g., a pollutant) dissolved in the fluid. Because of observational noise, and because there are cases where the fluid flow will not affect the movement of the pollutant, the fluid flow cannot be uniquely determined from the observations. We therefore adopt a statistical (Bayesian) approach, developing probabilistic estimates of the fluid flow using models of observation error and our understanding of the flow before measurements are taken. We provide conditions under which, as the number of observations grows large, the approach is able to identify the fluid flow that generated the observations. We define several efficient algorithms for computing statistics of the fluid flow, one of which involves approximating the movement of individual solute particles to estimate concentrations only where required by the inverse problem. We identify two interesting example problems for which the statistics of the fluid flow are very different. The first case produces an approximately normal distribution. The second example exhibits highly nonGaussian structure, where several different classes of fluid flows match the data very well. We use these examples to test the functionality and efficiency of several numerical (Markov Chain Monte Carlo) methods developed in recent years to compute the solution to similar problems.