The study of the magnetic properties of highly anisotropic paramagnetic molecules is an area of intense current research interest. Of these, single-molecule magnets (SMMs) and single-chain magnets (SCMs) showing non-equilibrium magnetization have remained a key topic over the past two decades. The slow magnetization reversals found in SMMs and SCMs are contingent on two requirements: a large ground-state spin forbidding direct quantum transitions of spin reversal, and a series of excited spin levels, due to the anisotropy of the system, which can act as steppingstones for the thermal relaxation of the spin orientations (the Orbach process). In this critical review, the latter requirement, i.e. the existence of magnetic anisotropies in paramagnetic species, is reviewed with the aim of providing clues towards the rational design of molecule-based magnets (100 references).