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This monograph explores de Branges' Hilbert space theory for entire functions and its application to spectral analysis of stationary processes and processes with stationary increments. Chapters examine the interplay between two areas of mathematics: functional analysis and probability theory. These distinct areas are made accessible to readers from either background by establishing the necessary analytic framework in Parts I and II before showing how they apply to Gaussian stochastic processes in Part III. Part I introduces fundamental notions and facts from the complex function theory on a half-plane, including de Branges functions, and Part II focuses on de Branges' theory. Part III covers spectral analysis of random processes, providing a deeper analysis of moving average representation of processes with stationary increments, their series expansion, and more. De Branges' Theory for Processes With Stationary Increments will be of interest to researchers in function theory as well as stochastic processes.