We review recent work on the modelling of atomic nuclei as quantized Skyrmions, using Skyrme's original model with pion fields only. Skyrmions are topological soliton solutions whose conserved topological charge B is identified with the baryon number of a nucleus. Apart from an energy and length scale, the Skyrme model has just one dimensionless parameter m proportional to the pion mass. It has been found that a good fit to experimental nuclear data requires m to be of order 1. The Skyrmions for B up to 7 have been known for some time, and are qualitatively insensitive to whether m is zero or of order 1. However, for baryon numbers B = 8 and above, the Skyrmions have quite a compact structure for m of order 1, rather than the hollow polyhedral structure found when m = 0. One finds that for baryon numbers which are multiples of four, the Skyrmions are composed of B = 4 sub-units, as in the α-particle model of nuclei.