This thesis presents detailed information about the state of art concerning the search for solutions of linear difference equations. It especially tries to fill theoretical gaps and improve some of the existing algorithms. The first chapter provides some fundamental definitions and theorems that are used throughout the entire thesis. The second chapter gives an overview about the theory of linear difference operators (with polynomial resp. rational function coefficients). This includes facts about the solution space and a survey about important operations on linear difference operators. In the subsequent four chapters algorithms for finding polynomial, rational, hypergeometric and d'Alembertian solutions are presented and compared.