We have studied homogeneous cavitation in liquid nitrogen and normal liquid helium. We monitor the fluid content in a large number of independent mesopores with an ink-bottle shape, either when the fluid in the pores is quenched to a constant pressure or submitted to a pressure decreasing at a controlled rate. For both fluids, we show that, close enough to their critical point, the cavitation pressure threshold is in good agreement with the Classical Nucleation Theory (CNT). In contrast, at lower temperatures, deviations are observed, consistent with a reduction of the surface tension for bubbles smaller than two nanometers in radius. For nitrogen, we could accurately measure the nucleation rate as a function of the liquid pressure down to the triple point, where the critical bubble radius is about one nanometer. We find that CNT still holds, provided that the curvature dependence of the surface tension is taken into account. Furthermore, we evaluate the first- and second-order corrections in curvature, which are in reasonable agreement with recent calculations for a Lennard-Jones fluid.