This paper is devoted to the study of composition operators on \(\sigma\)- finite standard Borel spaces. The first paragraph generalizes some ancient results concerning the characterization of Fredholm composition operators. The following two sections are dedicated to composition operators with ascent 1 and descent 1; a criterion for establishing partial isometry and co-isometry of operators is given. The last paragraph is concerned with hyponormal composition operators on square summable weighted standard Borel spaces.