This thesis is about constructions of canonical metrics in complex non-Kähler geome- try, focusing in particular on balanced metrics satisfying special hermitian curvature con- ditions.

More specifically, we adapt gluing strategies from Kähler geometry to obtain families of balanced metrics with special curvature properties, with particular relevance for the con- structions of solutions of the Hull-Strominger system and the geometrization of balanced classes. More specifically, we show that: crepant resolutions of orbifolds with isolated sin- gularities admitting singular Chern-Ricci flat balanced metrics can also be endowed with Chern-Ricci flat balanced metrics; small resolutions of smoothable Calabi-Yau singular threefolds with a finite family of Ordinary Double Points admit an approximately Chern- Ricci flat balanced metric; the blowup at a finite family of points of a compact Chern-Ricci flat balanced manifold always admits Chern-scalar constant balanced metrics. In all three cases we have a control on the Bott-Chern cohomology class of metrics constructed.

Furthermore, we use representation theory techniques to construct special balanced metrics on the class of real simple Lie groups of inner type, as well as on the correspond- ing compact homogeneous spaces, on which we obtain that the metrics constructed are Chern-scalar with non-vanishing Chern-Ricci tensor, providing a family of compact com- plex manifolds with vanishing first Chern class and non-vanishing first Bott-Chern class. Moreover, we show that for this class of homogeneous spaces the Fino-Vezzoni conjecture holds.