THE superconducting state of a material can be suppressed by either increasing the temperature (T) or applying a magnetic field (H). For bulk samples, the form of the H-T phase boundary is mainly determined by the material itself; sample topology can be neglected because the surface-to-volume ratio is small(1). But for mesoscopic samples, this ratio becomes very large and nucleation of the superconducting state should depend strongly on the boundary conditions imposed by the sample shape, analogous to the role of the confining potential on the energy levels in the quantum-mechanical ’particle-in-a-box’ problem(2). Here we describe measurements of the superconducting H-T phase boundary of a range of mesoscopic aluminium structures (lines, squares and square rings) which show clearly the effect of sample topology. The H-T phase boundaries determined experimentally are in excellent agreement with theoretical calculations.