The surface magneto-optic Kerr effect (SMOKE) was used to investigate the magnetic properties of epitaxial thin films of Co, Ni, and their alloys grown on Cu(100) and Cu(111). The Curie temperature T(C) is higher for the same films of a given thickness on Cu(111) than on Cu(100). All the films show a change in the power-law exponent beta of the magnetization density M approximately (1-T/T(C))beta with reducing film thickness. Ni films on Cu(100) undergo a particularly abrupt crossover at approximately 7 monolayers (ML) from three-dimensional Heisenberg (beta=0.37) to finite-size two-dimensional XY (beta=0.23) behavior as the film thickness is reduced. The characteristic power-law exponent beta=0.23 of these films appears to be an experimental realization of Kosterlitz-Thouless behavior over a restricted temperature range. A similar, but more gradual crossover occurs for the Ni films on Cu(111) at 8 to 12 ML. The finite-size scaling behavior in the few-monolayers-thickness range is compared with that reported for Ising thin-film behavior. In all instances T(C) extrapolates with decreasing thickness to zero at one monolayer. The dimensionality crossover and finite-size scaling behavior is discussed in the light of our current understanding of spin-wave quantization, anisotropy, and film microstructure.