A description of thermodynamics for continuum mechanical systems is presented in the coordinate-free language of differential geometry. First, a careful description of the mathematical tools that are needed to formulate the relevant conservation laws is given. Second, following an axiomatic approach, the two thermodynamic principles will be described, leading to a consistent description of entropy creation mechanisms on manifolds. Third, a specialisation to Fourier-Navier-Stokes fluids will be carried through.