Multiboson spin waves are constructed to represent the low-temperature collective states of paramagnets and ferromagnets whose spins are located in strong single-ion anisotropy fields and are coupled by weak exchange interactions. The different sets of bosons are particles excitable to the different eigenstates of the single-ion part of the Hamiltonian, and boson representations of the single-ion spin operators are constructed by a matrix-elements-matching method. The method allows the determination of magnon frequencies and of possible effects of applied oscillatory magnetic fields, and also allows the estimation of boundaries between different magnetic phases. However, it does not establish the form of the single-ion contribution to four-magnon interactions. In addition, it is not applicable to transitional phases where the exchange cannot be treated as a perturbation. The spin-1 system with positive and negative uniaxial and orthorhombic anisotropies in various magnitudes of parallel static magnetic fields is discussed in detail. Results of Ishikawa and Oguchi are obtained and generalized. The presence of orthorhombic anisotropy is predicted to make possible the "parallel" pumping of various magnon pairs, and also to give rise to parallel incoherent resonance absorption between excited states, as well as the usual "perpendicular" coherent $k=0$ ground-state resonance excitation. Magnon relaxation times are extimated in the case of a paramagnet with hard-axis anisotropy in sufficiently small magnetic fields. Typical materials to which this theory applies are hydrated nickel salts.