Let Xin = {1, 2, ..., n}. In this thesis we obtain some combinatorial results in the semigroups of full and partial transformations of Xin complementing the results of Umar and Laradji. We compute the cardinalities of some equivalence classes defined by different parameters in the semigroups of full and partial transformations of Xn and recover some results from literature review in these semigroups. Here we also investigate the sets of idempotents in the full and partial transformations of Xn and obtain some combinatorial results. We compute the cardinalities of some equivalence classes defined by different parameters in these sets. Moreover, we obtain some recurrence relations and exponential generating functions of the number of idempotents in these semigroups.