A wide class of problems from medical imaging, robotics, astrophysics, economics, etc. can be formulated as inverse problems. Solving such problems generally starts by the so-called direct or forward modelling that theoretically describes how input parameters x are translated into effects y. Then from experimental observations of these effects, the goal is to find the parameter values that best explain the observed measures. Typical situations and constraints that can be encountered in practice are that 1) both direct and inverse relationships are highly non-linear; 2) the observations y are high-dimensional (eg. signals in time or spectra); 3) many such high-dimensional observations are available and the application requires a very large number of inversions; 4) the parameters x to be predicted is itself multi-dimensional with correlated dimensions. In addition, it is common that direct models are available only through mechanistic formulations that provide high-fidelity simulations of the system but only through a “black-box” poorly suited for statistical inference. The main challenge comes from the fact that the model likelihood is typically intractable and has to be estimated. These situations are referred to as likelihood-free or simulation-based inference and have received a lot of attention in recent years with momentum coming from mixing ideas from statistics and machine learning. The proposed approach is illustrated in neuroimaging and in particular with the recent concept of Magnetic Resonance Fingerprinting.