The main result of this thesis is a classification of all homogeneous spaces G/H admitting a G-invariant G2 -structure, assuming that G is a connected compact Lie group and G acts effectively on G/H. They include a subclass of all homogeneous ̃ spaces G/H with a G-invariant G2 -structure, where G is a compact Lie group. There are many new examples with nontrivial fundamental group. A formula computing the ̃ dimension of the space of G-invariant structures (resp. of G-invariant G2 -structures) on G/H is given. We study a subclass of homogeneous spaces of high rigidity and low rigidity and show that they admit families of invariant co-closed G2 -structures (resp. ̃ ̃ G2 -structures). Some new interesting examples of G2 -structures on these spaces are ̃ found. We also present a scheme of classification of G2 -structures using their intrinsic torsion.