A mesopump having a plurality of elementary cells and a method of pumping fluids using the mesopump. The cells each have a body forming a cavity having upper and lower electrodes with curved surfaces facing the other electrode. Electrical connections selectively energize the electrodes. A bistable diaphragm is mounted with its major portion located in the cavity between the electrodes so that the diaphragm deflects toward the upper or lower surfaces to conform thereto. A lateral conduit is located at at least one end of the body for fluid communication with the cavity. The lateral conduits are operably connected to a portion of the diaphragm mounted in the body and are opened and closed by movement of the diaphragm. A vertical conduit on at least one curved surface controls flow of fluid by movement of the diaphragm into and out of contact with the vertical conduit. The cells are interconnected to form the mesopump whereby activation of the electrodes causes movement of the diaphragm between the electrodes to move fluid into and out of the body. The preferred mesopump includes sets of three cell groups so that fluid enters a first cell from a source of fluid, is expelled to a second cell through a vertical conduit; expelled via a lateral conduit to a third cell, and expelled therefrom, whereby fluid passes from cell to cell upon activation of the electrodes.