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$G_2$-structures defined by a closed positive 3-form constitute the starting point in known and potentially effective methods to obtain holonomy $G_2$ 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 $G_2$-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 $G_2$-structures. This is based on joint works with F. Podestà   (Firenze), A. Fino (Torino), and M. Fernà ¡ndez (Bilbao).