We present our most recent results regarding near-BPS Skyrmions and argue that they provide an improved description of nucleons and nuclei. For some years now, the Skyrme Model has been considered a natural candidate for a low-energy effective theory of QCD, a point of view supported by results coming from 1/N expansion and holographic QCD. This framework leads to an attractive picture where baryons (and nuclei) emerge as topological solitons with a topological number identified to the baryon number. But even the most naive Skyrme Model extensions have been plagued with the same problem: they predict large binding energies for the nuclei. On the other hand, the solutions that arise from the more recently proposed near-BPS Skyrme model nearly saturate the Bogomol'nyi bound which means that by construction they must have small binding energies. We address here the issue regarding the energy minimizer which remains unknown for A > 1 by proposing a more appropriate ansatz than the usual axially symmetric solution.