We provide an extensive overview of upper bounds on the number of gates needed in reversible and quantum circuits. As reversible gate libraries we consider single-target gates, mixed-polarity multiple-controlled Toffoli gates, and the set consisting of the NOT, the CNOT, and the two-controlled Toffoli gate. As quantum gate libraries we consider the semi-classical NCV library (consisting of NOT, CNOT, and the square-root of NOT called V ) as well as the universal and commonly used Clifford+TClifford+T gate library. Besides a summary of known bounds, the paper provides several new and tighter bounds. Several synthesis approaches and mapping schemes were used to calculate the bounds.