Magnetic nanoparticle hyperthermia and thermal ablation have been actively studied experimentally and theoretically. In this review, we provide a summary of the literature describing the properties of nanometer-scale magnetic materials suspended in biocompatible fluids and their interactions with external magnetic fields. Summarised are the properties and mechanisms understood to be responsible for magnetic heating, and the models developed to understand the behaviour of single-domain magnets exposed to alternating magnetic fields. Linear response theory and its assumptions have provided a useful beginning point; however, its limitations are apparent when nanoparticle heating is measured over a wide range of magnetic fields. Well-developed models (e.g. for magnetisation reversal mechanisms and pseudo-single domain formation) available from other fields of research are explored. Some of the methods described include effects of moment relaxation, anisotropy, nanoparticle and moment rotation mechanisms, interactions and collective behaviour, which have been experimentally identified to be important. Here, we will discuss the implicit assumptions underlying these analytical models and their relevance to experiments. Numerical simulations will be discussed as an alternative to these simple analytical models, including their applicability to experimental data. Finally, guidelines for the design of optimal magnetic nanoparticles will be presented.