It is well-known that the existence of a Kähler metric on a compact manifold imposes strong topological obstructions. In contrast, by a result of Gauduchon, in the conformal class of any given Hermitian metric on a compact complex manifold X there always exists a metric satisfying the condition $\partial \bar\partial F^{n-1}=0$, where F denotes the fundamental form and n is the complex dimension of X.

Between the Kähler class and the Gauduchon class, other interesting classes of special Hermitian metrics have been considered in relation to several problems in differential and algebraic geometry.

In this talk we focus on holomorphic deformations of compact complex manifolds endowed with balanced or strongly Gauduchon metrics.