For a graph G, the minimum dominating set problem is to find a minimum size set S of vertices of G such that every vertex is either in S or adjacent to a vertex in the set. The decision version of this problem, which asks whether G has a dominating set of a particular size k, is known to be NP-complete, and no polynomial time algorithm to solve the problem is currently known to exist. The queen domination problem is to find the minimum number of queens which, collectively, can attack every square on an n by n chess board. The related border queen problem is to find such a collection of queens with the added restriction that all queens lie on the outer border of the board. This thesis studies practical exponential time algorithms for solving domination problems, and presents an experimental comparison of several different algorithms, with the goal of producing a broadly effective general domination solver for use by future researchers. The developed algorithms are then used to solve several open problems, including cases of the queen domination problem and the border queen problem. In addition, new theoretical upper bounds are presented for the border queen problem for some families of queen graphs. / Graduate