The well-known Landau's theorem states that, for any positive integer $k$, there are finitely many isomorphism classes of finite groups with exactly $k$ conjugacy classes. We will discuss some variations of this theorem for $p$-regular classes as well as $p$-singular classes. We will present several results showing that the structure of a finite group $G$ is strongly restricted by the number of $p$-regular classes or the number of $p$-singular classes of $G$. This is a joint work with Alexander Moreto.