G2-structures defined by a closed positive 3-form constitute the starting point in known and potentially effective methods to obtain holonomy G2 metrics on seven-dimensional manifolds. Currently, no general results guaranteeing the existence of such structures on compact 7-manifolds are known. Moreover, the construction of new explicit examples requires substantial efforts. In the first part of this talk, I will discuss the properties of the automorphism group of a compact 7-manifold endowed with a closed G2-structure, showing how they impose strong constraints on the construction of examples with high degree of symmetry (e.g. homogeneous, cohomogeneity one). Then, I will focus on the non-compact case. Here, motivated by known results on symplectic Lie groups, I will discuss some structure theorems for Lie groups admitting left invariant closed G2-structures. This is based on joint works with F. Podestà (Firenze), A. Fino (Torino), and M. Fernández (Bilbao).