Non-integrable special geometric structures in dimensions six and seven

Six-dimensional manifolds admitting an SU(3)-structure and seven-dimensional manifolds endowed with a G2-structure are the main object of study in this thesis. In the six-dimensional case, we consider SU(3)-structures (ω,ψ) satisfying the condition dω = c ψ, c ∈ R − {0}, known in literature as coupled. They are half-flat and generalize the class of nearly Kähler SU(3)-structures. We study their properties in the general case and in relation with the rôle they play in supersymmetric string theory, the conditions under which the associated metric is Einstein, their behaviour with respect to the Hitchin flow equations and various classes of examples. In the seven-dimensional case, we focus on G2-structures defined by a stable 3-form φ which is locally conformal equivalent to a closed one. We study the restrictions arising when the underlying metric is Einstein, we use warped products and the mapping torus construction to provide noncompact and compact examples of 7-manifolds endowed with such a structure starting from 6-manifolds with a coupled SU(3)-structure and, finally, we prove a structure result for compact 7-manifolds. We conclude studying a generalization of the Hitchin flow equations and a geometric flow of spinors on 6-manifolds. The latter gives rise to a flow of SU(3)-structures.