Our theoretical study of magnetic nanoparticles is based on an original analytic approach, which allows to track down the intrinsic mechanisms of the observed magnetic properties. In particular, it reveals some misleading aspects of the standard phenomenological description of ferromagnetic nanoparticles. We show that quantum effects become important for particle size below 50 nm even at high temperatures. Several new results are relevant for controlling magnetization in nanostructures. Thus, the theory predicts that dividing a nanoparticle in two pieces may enhance and even generate a spontaneous magnetization. The effect grows nonlinearly with smaller size and is especially large for structures with anisotropic shape. Generally, at a given temperature and given number of atoms nanoparticles of a more isotropic shape have a larger polarization. However, coupling to environment strongly affects the magnetization: for a free-standing particle it is progressively suppressed for smaller sizes while for a particle with surface spins pinned by the coupling this trend is reversed. Due to this boundary induced polarization mechanism the Curie temperature Tc in the latter case may be several times larger. Moreover, anisotropic structures are much more sensitive to the environment than the isotropic ones: e.g., placing a free-standing nanorod into an appropriate environment may lead to an abrupt increase in its magnetization, which can be larger than that of a cubic particle of the same volume.