In this paper a unified probabilistic framework for solving inverse problems in the presence of epistemic and aleatory uncertainty is presented. The aim is to establish a flexible theory that facilitates Bayesian data analysis in experimental scenarios as they are commonly met in engineering practice. Problems are addressed where learning about unobservable inputs of a forward model, e.g. reducing the epistemic uncertainty of fixed yet unknown parameters and/or quantifying the aleatory uncertainty of variable inputs, is based on processing response measurements. Approaches to Bayesian inversion, hierarchical modeling and uncertainty quantification are combined into a generic framework that eventually allows to interpret and accomplish this task as multilevel model calibration. A joint problem formulation, where quantities that are not of particular interest are marginalized out from a joint posterior distribution, or an intrinsically marginal formulation, which is based on an integrated likelihood function, can be chosen according to the inferential objective and computational convenience. Fully Bayesian probabilistic inversion, i.e. the inference the variability of unobservable model inputs across a number of experiments, is derived as a special case of multilevel inversion. Borrowing strength, i.e. the optimal estimation of experiment-specific unknown forward model inputs, is introduced as a means for combining information in inverse problems. Two related statistical models for situations involving finite or zero model/measurement error are devised. Multilevel-specific obstacles to Bayesian posterior computation via Markov chain Monte Carlo are discussed. The inferential machinery of Bayesian multilevel model calibration and its underlying flow of information are studied on the basis of a system from the domain of civil engineering. A population of identically manufactured structural elements serves as an exemplary system for examining different experimental settings from the standpoint of uncertainty quantification and reduction. In a series of tests the material variability throughout the ensemble of specimens, the entirety of specimen-specific material properties and the measurement error level are inferred under various uncertainties in the problem setup.