The effects of growth rate (Gr), deposition temperature (Td), film thickness (tF), and substrate-induced strain (Є) on morphological, crystallographic, and magnetic characteristics of equiatomic CoPt epitaxial films synthesized with pulsed laser deposition are investigated. The (001)-oriented single-crystal substrates of MgO, SrTiO3, and LaAlO3 provide different degrees of epitaxial strain for growth of the disordered face-centered cubic (fcc) and ordered face-centered tetragonal (L10) phases of CoPt. The films deposited at Td≈600 °C on all three substrates are fcc with in-plane magnetization and a narrow hysteresis loop of width ≈200 Oe. The L10 phase, stabilized only at Td≥700 °C, becomes predominantly c-axis oriented as Td is increased to 800 °C. While the crystallographic structure of the films depends solely on the Td, their microstructure and magnetization characteristics are decided by the growth rate. At the higher Gr(≈1 Å/s) the L10 films have a maze-like structure which converts to a continuous film as the tF is increased from 20 to 50 nm. The magnetic coercivity of these films increases as the L10 phase fraction grows with Td and its orientation becomes out of- the film plane. The evolution of microstructure with Td is remarkably different at a lower growth rate (≈0.4 Å/s). Here, the structure changes from a self-similar fractal pattern to a disordered assembly of nanodots as the Td is raised from 700 to 800 °C, and is understood in terms of the imbalance between strain and interfacial energies. Magnetic force microscopy of such films reveals no distinct domain walls within the nanoislands, while a clear contrast is seen between the islands of reversed magnetization. Magnetic relaxation measurements on these assemblies of single-domain islands show a negligible decay of magnetization unless a reverse field close to the coercive field (Hc≈30 kOe) is applied. The simple picture of coherent rotation of moment appears incompatible with the time dependence of the remanent magnetization in these films.