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For a compact oriented surface $\Sigma$ of genus $g$ with $n+1$ boundarycomponents, the space $\mathfrak{g}(\Sigma)$ spanned by free homotopy classesof loops in $\Sigma$ carries the structure of a Lie bialgebra equipped with anatural decreasing filtration, whose structure morphisms are called the Goldmanbracket and the (framed) Turaev cobracket.  We address the following Goldman-Turaev (GT) formality problem: construct aLie bialgebra homomorphism $\theta$ from $\mathfrak{g}(\Sigma)$ to itsassociated graded ${\rm gr}\, \mathfrak{g}(\Sigma)$ such that ${\rm gr} \,\theta = {\rm id}$. In order to solve it, we define a family of higher genusKashiwara-Vergne (KV) problems for an element $F\in {\rm Aut}(L)$, where $L$ isa free Lie algebra. In the case of $g=0$ and $n=2$, it is the classical KVproblem from Lie theory. For $g>0$, these KV problems are new.  We show that an element $F$ induces a GT formality map if and only if it is asolution of the KV problem. A crucial step in solving the higher genus KVproblem is to construct solutions for the case of $g=1$ and $n=1$ in terms ofcertain elliptic associators following Enriquez. By solving the KV problem, weestablish the GT formality for every $g$ and $n$, with the exception of someframings for $g=1$ in which case the GT formality actually does not hold.Furthermore, we introduce pro-unipotent groups ${\rm KV}$ and ${\rm KRV}$ whichact on the space of solutions of the KV problem freely and transitively. Thereare injective maps ${\rm GT}_1\to {\rm KV}, {\rm GRT}_1\to {\rm KRV}$ fromGrothendieck-Teichm\"uller groups.  As an application, we show that the Johnson obstruction given by the Turaevcobracket coincides with the one given by the Enomoto-Satoh trace. As part ofour study, we prove a uniqueness theorem for non-commutative divergencecocycles on the group algebra of a free group which is of independent value.

The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera