In this chapter, we introduce a category theory-based conceptual modelling framework, that formally describes physics-based models appearing in engineering practice. At first, a categorical description of mathematical models is introduced, and their properties are discussed. After that, coupled models are addressed by lifting the results from categories of mathematical models to the categories of coupled models. However, because of the coupling, the structure of categories of coupled models is more complicated. In particular, it is necessary to distinguish if models from the same or from different categories are coupled. Finally, more abstract categorical constructions on categories of mathematical models are discussed. For supporting the reader, the chapter is provided with various illustrative examples for practical interpretations of the abstract constructions introduced in this chapter.