A key question in the theory of Markov chains and stochastic matrices is the existence of stochastic c-th roots: given a stochastic matrix A, can we find another stochastic matrix B such that A equals B raised to the power of c, for some integer c in the natural numbers? This problem is closely tied to the modeling of Markov chain processes, where a stochastic c-th root of a transition matrix A for a time period t_0 serves as the transition matrix for a shorter time period t_0 divided by c. In this thesis, we explore two contrasting scenarios for stochastic matrices A and B such that A equals B raised to the power of c, where c is a natural number: (1) when c can take infinitely many values, and (2) when A belongs to a very restrictive subset of stochastic matrices and solutions B exist only for finitely many c. In the first case, if A has a stochastic c-th root for every c in the natural numbers, the matrix is termed infinitely divisible. Additionally, if A is nonsingular, the associated Markov chain is said to be embeddable. Extending this framework, we introduce "arbitrarily finely divisible" stochastic matrices, abbreviated as AFD matrices, which have stochastic roots for infinitely many, but not necessarily all, natural numbers c. The significance of AFD matrices lies in their ability to provide transition matrices for the associated Markov process over infinitesimally short time intervals, even when the matrix is not embeddable. This thesis establishes the foundation for the study of AFD matrices by presenting a collection of general results on the arbitrary fine divisibility of matrices (complex, real, and stochastic) and their application to specific cases: 2 by 2 stochastic matrices, 3 by 3 circulant stochastic matrices, and characterization of irreducible rank-two AFD matrices. The spectral properties of stochastic matrices are tightly connected with the Karpelevic region, denoted Theta_n, which is the region in the complex plane containing all the eigenvalues of all n by n stochastic matrices. The boundary of this region, denoted as partial Theta_n, is composed of arcs known as Karpelevic arcs. The thesis provides a complete characterization of Karpelevic arcs that can be expressed as a power of another Karpelevic arc, resolving an open problem for which only partial solutions were previously available. This result not only refines the necessary conditions for the existence of stochastic roots for matrices whose eigenvalues lie on the boundary of Theta_n but also imposes restrictions on the allowed values of c in the natural numbers in the equation A equals B raised to the power of c, where A and B are matrices realizing eigenvalues on the boundary of Theta_n. With the above restriction on c in the natural numbers for A equals B raised to the power of c, where A and B realize eigenvalues on the boundary of Theta_n, the thesis provides a full characterization of the sparsest matrices realizing eigenvalues on the boundary of the Karpelevic region that possess a c-th root. This is achieved by studying the powers of the strongly connected digraphs of these sparsest realizing matrices.