Flow matching algorithms and their variants have recently emerged as powerful tools for the generation of complex data modalities, including text, images, and videos, conditioned on user-provided inputs. Beyond their success in generative modeling, these methods have begun to play an important role in probabilistic machine learning, particularly for addressing stochastic forward and inverse problems that arise in scientific and engineering applications. In this talk, we focus on their application to the solution of Bayesian inverse problems, where the objective is to infer unknown parameters or fields from indirect and noisy observations through a probabilistic framework. We will discuss several key properties that make flow matching methods especially attractive in this context. These include their ability to incorporate highly complex prior distributions that may be available only through samples rather than in closed analytic form, their capacity to interface with sophisticated forward models treated as black-box simulators, their natural compatibility with non-Gaussian prior and posterior distributions, and their scalability to high-dimensional parameter spaces, which is essential for practical inverse problems governed by partial differential equations. In addition, we will present theoretical and methodological questions and motivate further analysis. We will examine the convergence behavior of the learned flows as the amount of training data and optimization time increase and discuss conditions under which the resulting approximations can be expected to faithfully represent the true Bayesian posterior. We will also explore connections with optimal transport theory, focusing on the transport of conditional probability densities and how these ideas can be leveraged to design more efficient sampling strategies and accelerate posterior exploration. We will demonstrate these developments through numerical experiments that include both canonical benchmark inverse problems, and more complex real-world applications that demonstrate the practical potential of flow matching approaches.