Certain properties that are common to all finite transformation semigroups are discussed. For example special properties of ideals in transformation semigroups are established. It is also proved that every element of a finite transformation semigroup must be one-to-one from some maximal subset of its domain onto that same set. This maximal subset is decomposed into cycles, and results are obtained connecting the orders of the cycles of an element and the order of the monogenic semigroup generated by that element. Numerical results concerning arbitrary subsemigroups in the transformation semigroup on three elements are listed. (Author).