Domain decomposition methods are powerful iterative methods for solving systems of algebraic equations arising from the discretization of partial differential equations by, e.g., finite elements. The computational domain is decomposed into overlapping or nonoverlapping subdomains. The problem is divided into, or assembled from, smaller subproblems corresponding to these subdomains. In this dissertation, we focus on domain decomposition methods for mortar finite elements, which are nonconforming finite element methods that allow for a geometrically nonconforming decomposition of the computational domain into subregions and for the optimal coupling of different variational approximations in different subregions. We introduce a FETI method for mortar finite elements, and provide numer- ical comparisons of FETI algorithms for mortar finite elements when different preconditioners, given in the FETI literature, are considered. We also analyze the complexity of the preconditioners for the three dimensional versions of the algorithms. We formulate a variant of the balancing method for mortar finite elements, which uses extended local regions to account for the nonmortar sides of the subre- gions. We prove a polylogarithmic condition number estimate for our algorithm in the geometrically nonconforming case. Our estimate is similar to those for other Neumann{Neumann and substructuring methods for mortar finite elements. In addition, we establish several fundamental properties of mortar finite elements: the existence of the nonmortar partition of any interface, the L^2 stability of the mortar projection for arbitrary meshes on the nonmortar side, and prove Friedrichs and Poincare inequalities for geometrically nonconforming mortar elements.