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In this thesis, a class of combinatorial optimization methods rooted in statistical mechanics and their use in signal processing applications will be discussed. The thesis consists of two separate parts. The first part deals with the rationale for my work and also covers the background information necessary to put the second part, which consists of a number of papers, in context. There are (at least) two sides to an optimization problem---the problem statement arising from an application or a design and the selection of an algorithm to solve the problem. In this work the problem statements are practical problems, of combinatorial nature, frequently encountered in signal processing and the algorithms of choice are annealing based algorithms, founded in statistical mechanics. From my work, it is my experience that solving a particular problem often leads to new developments on the part of the algorithm which, in turn, open up possibilities to apply the modified algorithm to a new set of problems, leading to a continuously improving algorithm and a growing field of applications. The included papers deal with the application of annealing optimization methods to the problems of configuring active noise and vibration control systems, digital filter design and adaptive filtering. They also describe the successive development of a highly efficient entropy-directed deterministic annealing (EDDA) optimization algorithm detailed in the final paper.