Let X be a minimal complex projective variety. Over the past years, many similar inequalities between the Chern classes of X have been obtained. Moreover, it is known precisely which varieties X can achieve the equality. However, so far all results in this direction have focussed on the case where the numerical dimension of X is either very small or very large. In this talk, I will present analogous inequalities for varieties of intermediate Kodaira dimension and I will present a characterisation of those varieties achieving the equality. This talk is partially based on joint work with Masataka Iwai and Shin-ichi Matsumura.