The effects of interactions and anisotropy on the magnetic properties of linear chains of superparamagnetic nanoparticles are studied theoretically by mapping the problem onto spin models. With zero anisotropy, the magnetic dipole moments are free to rotate, and the system resembles a classical ferromagnetic Heisenberg model with long-range dipolar interactions. With strong anisotropy, they are constrained to align with the chain, and the system resembles the classical ferromagnetic Ising model with long-range interactions. Using a modified mean-field theory, expressions for the magnetization curve and initial magnetic susceptibility are derived from the response of a single particle subject to an effective field arising from the applied field and the interactions with the other particles. Various approximations for the effective field are tested against results from Monte Carlo simulations. It is shown that, for physically relevant interaction strengths, reliable theoretical predictions for both the zero-anisotropy and strong-anisotropy cases can be derived in a simple closed form.