In this paper, for the first time a design for a Reversible Programmable Logic Array (RPLA) is introduced. This is the first RPLA design because the reversible PLA design, presented in previous research, is not a programmable circuit. In our presented RPLAs, four reversible AND array designs with different specifications are proposed. A reversible OR array, which can be programmed to generate any Boolean function, is also designed. This reversible and programmable OR array is also cascadable. That is, it produces a copy of inputted midterms at its outputs to be fed to another OR array in order to design another Boolean function, with no need for another AND array. The proposed 3-input RPLA is programmed to design three reversible circuits, a 1-bit full adder, a 1-bit full subtractor, and a 2-to-1 line multiplexer. Five figures of merit, including number of gates, number of constant inputs, number of garbage outputs, depth and quantum cost of the circuit are considered to evaluate and compare the designs. A comparison between the proposed reversible AND arrays and the same circuit presented in previous research, against these figures of merit, shows a better performance of our proposed designs. The proposed RPLAs are also evaluated, using these five figures of merit.