Introduction

The weak normal covering number and the normal covering number of the finite non-abelian simple groups. The normal 2-coverings and the weak normal covering number of the almost simple groups. Some applications: the invariably generating graph and the Aut-invariably generating graph; the Erdős–Ko–Rado theorem and the derangement graph; the Boston–Shalev conjecture; normal 2-coverings for arbitrary finite groups; normal coverings and Kronecker classes.