Abstract: 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  focused 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.