  We define a family ${\rm KV}^{(g,n)}$ of Kashiwara-Vergne problems associated
with compact connected oriented 2-manifolds of genus $g$ with $n+1$ boundary
components. The problem ${\rm KV}^{(0,3)}$ is the classical Kashiwara-Vergne
problem from Lie theory. We show the existence of solutions of ${\rm
KV}^{(g,n)}$ for arbitrary $g$ and $n$. The key point is the solution of ${\rm
KV}^{(1,1)}$ based on the results by B. Enriquez on elliptic associators. Our
construction is motivated by applications to the formality problem for the
Goldman-Turaev Lie bialgebra $\mathfrak{g}^{(g, n+1)}$. In more detail, we show
that every solution of ${\rm KV}^{(g,n)}$ induces a Lie bialgebra isomorphism
between $\mathfrak{g}^{(g, n+1)}$ and its associated graded ${\rm gr} \,
\mathfrak{g}^{(g, n+1)}$. For $g=0$, a similar result was obtained by G.
Massuyeau using the Kontsevich integral.
  This paper is a summary of our results. Details and proofs will appear
elsewhere.
