In the field of ultracold atoms, two major recent research directions have been rapidly rotating Bose-Einstein condensates, and condensates loaded into optical lattice potentials. While past optical lattice work has concentrated largely on the regime of few correlated particles per lattice site, in this thesis the opposite limit of tunnel-coupled arrays of macroscopic condensates—Josephson junction arrays—is studied. Combining such Josephson junction arrays with rapidly rotating BECs yields highly complex quantum fluid systems with many analogies to condensed-matter systems. In particular, optical lattice potentials allow the creation of low-dimensional, or effectively low-dimensional systems. (1) In this thesis, the first rotating two-dimensional optical lattice potential for ultracold atoms was created, which served as a periodic pinning potential for vortices, and induced a structural crossover in the vortex lattice. (2) The Berezinskii-Kosterlitz-Thouless superfluid transition was studied in a non-rotating, finite-temperature, two-dimensional array of Josephson junctions. (3) Rapidly rotating BECs were loaded into a one-dimensional optical lattice aligned with the rotation axis. A crossover from a coherent array of Josephson-coupled rotating BECs, to an array of isolated, two-dimensional rotating BECs was observed, and the ensuing vortex lattice fluctuations were studied. (4) Studies of vortex lattices in rotating BECs were extended to two-component BECs, where a self-organization of vortices in the two components into two interlaced square vortex lattices was observed.