### Abstract:

Noncommutative rational functions form a free skew ﬁeld, a universal object in the category of skew ﬁelds. They emerge in various branches of pure and applied mathematics, such as noncom-mutative algebra, automata theory, control theory, free analysis, free real algebraic geometry and free probability. A noncommutative rational function is given by a formal rational expression involving freely noncommuting variables and arithmetic operations, which can be naturally evaluated at tuples of matrices of all sizes. This dissertation studies such ﬁnite-dimensional evaluations, with the goal of determining what information about the structure of the free skew ﬁeld can be recovered from them. Firstly a matrix coeﬃcient realization theory is developed, which for an arbitrary noncom-mutative rational function yields an eﬃciently computable normal form. Such a realization of a noncommutative rational function measures its complexity and describes its natural domain as the complement of a free singularity locus of a linear matrix pencil. The inclusion problem for free loci, and thus for domains of noncommutative rational functions, is next solved in terms of epimorphisms between the coeﬃcient algebras of the corresponding linear matrix pencils. This theorem, called a Singularitatstellensatz for linear matrix pencils, is closely related to the invariant theory of the general linear group. Via realization theory this result yields an algebraic characterization of noncommutative rational functions with a given domain. More-over, a description of noncommutative rational functions whose domains contain all tuples of hermitian matrices is derived. In particular, it is proven that an everywhere deﬁned noncom-mutative rational function is a noncommutative polynomial. The understanding of the behavior of noncommutative rational functions under hermitian evaluations is further advanced by the resolution of an noncommutative analog of Hilbert’s 17th problem: a noncommutative rational function that is positive semideﬁnite at every tuple of hermitian matrices is a sum of hermitian squares of noncommutative rational functions. Finally, the construction of the free skew ﬁeld via matrix evaluations is extended to partially commuting arguments. The obtained multipartite rational functions play a remarkable role in the theory of universal skew ﬁelds of fractions and in the diﬀerence-diﬀerential calculus in free analysis.