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This book marks a milestone in the theory and algorithms of the Schroedinger bridge and its generalizations-a topic of significant current interest across statistical mechanics, stochastic control theory, and generative AI. The topic had its genesis in physics, specifically in 1931 when Erwin Schroedinger conceived a thought experiment to interpret quantum mechanics through the lens of non-equilibrium Markov processes. Schroedinger's idea was much ahead of its time: neither the axiomatic theory of stochastic processes, nor the mathematics of large deviations principle were available then. The connections with stochastic control became clear only in the late twentieth century. In the current time, the development of diffusion models in generative AI have re-energized this area via vigorous interdisciplinary research. Simply put, Schroedinger bridge is a diffusion process that interpolates probability distributions observed at given times with maximum likelihood guarantee. This book contributes to two significant generalizations of the Schroedinger bridge: one where prior knowledge in underlying physics is incorporated as some prescribed drift and diffusion coefficients, the other where additional regularization is enforced on the paths of the diffusion process. For the former, this book establishes contraction guarantees for convergence of numerical algorithms. For the latter, this book shows how regularization manifests as reaction rate and derives corresponding Markov kernels for nontrivial models of practical interest. New points of contact with quantum mechanics, in the spirit of Schroedinger's original work, are also uncovered.