We consider the general hypermultiplet low-energy effective actions (LEEA) that may appear in four-dimensional, N=2 supersymmetric Yang-Mills theories, e.g. in the Coulomb and Higgs branches. Our main purpose is a description of the exact LEEA of n magnetically charged hypermultiplets. The hypermultiplet LEEA is given by the N=2 supersymmetric non-linear sigma-model (NLSM) with a 4n-dimensional hyper-Kähler metric subject to relevant isometries. The harmonic superspace (HSS) and NLSM isometries are used to constrain the hyper-Kähler potential of the metric in question. The N=2 supersymmetric projections of HSS superfields to N=2 linear (tensor) O(2) and O(4) multiplets in N=2 projective superspace (PSS) are used to deduce the explicit form of the hypermultiplet LEEA metric in some particular cases. As the by-product, a simple new classification of all multi-monopole moduli space metrics having SO(3) rotational symmetry is proposed in terms of real quartic polynomials of 2n variables, modulo Sp(n) transformations. The 4d hypermultiplet LEEA for n = 2 can be encoded in terms of an elliptic curve.