The normal/superconducting phase boundary T-c has been calculated for mesoscopic loops as a function of an applied perpendicular magnetic field H. While for thin-wire loops and filled disks, the T-c(H) curves are well known, the intermediate case, namely, mesoscopic loops of finite wire width, have been studied much less. The linearized first Ginzburg-Landau (GL) equation is solved with the proper normal/vacuum boundary conditions both at the internal and at the external loops radii. For thin-wire loops, the T-c(H) oscillations are perfectly periodic, and the T-c(H) background is parabolic (this is the usual Little-Parks effect). For loops of thicker wire width, there is a crossover magnetic field above which T-c(H) becomes quasi-linear, with the period identical to the T-c(H) of a filled disk (i.e., pseudoperiodic oscillations). This dimensional transition is similar to the 2D-3D transition for thin films in a parallel field, where vortices start penetrating the material as soon as the film thickness exceeds the temperature-dependent coherence length by a factor 1.8. For the presently studied loops, the crossover point is controlled by a similar condition. In the high-field '3D' regime, a giant vortex state establishes, where only a surface superconducting sheath near the sample's outer radius is present. (C) 2000 Elsevier Science B.V. All rights reserved.