Solving preserver problems represent an active research area in matrix theory. Tipically it demands a characterization of all maps between two given sets of matrices that fulfill certain additional ...criteria. When preserver problems are studied on matrices with coefficients from a finite field, a mixture of various mathematical subdisciplines can be used to attact the problem. These include graph theory, matrix theory, theory of finite fields, and finite geometry. Since these methods are not yet widely known and did not yet reach their fully potential, we survey them in this chapter along a list of selected results. To make the content accessible as much as possible, we include basic definitions and results from the mathematical areas mentioned above, which one encounters when dealing with preserver problems. In particular, the chapter contains: an introduction to preserver problems; normal forms of various types of matrices, with a special emphasis on matrices over a finite field; a descripption of the connection between preserver problems over finite fields and graph homomorphisms/chromatic graph theory; basic properties of several types of graphs that arise from various mathematical structures over a finite field, with an emphasis on their spectrum.
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