of the boundary relation $\G$ is defined by $$\begin{gathered}
M(\l)=\{\{\G_0\{f_\l,\l f_\l\}, \G_1\{f_\l,\l f_\l\}\}: f_\l\in\ker (A^*-\l)
\}, \quad\l\in\CR\end{gathered}$$ and now it belongs to the class of Nevanlinna families; moreover, if the map $\G_0$ is surjective, then $M(\l)$ is a Nevanlinna operator function. In the paper [@DM09] the Weyl function was used for description of various classes of the exit space extensions $\wt A(=\wt A^*)\supset A$.

In the present paper the new concept of a boundary relation for $A^*$ with possibly unequal boundary spaces $\cH_0$ and $\cH_1$ is introduced. Roughly speaking this relation is a (possibly multivalued) linear map $\G:=(\G_0\;\;\G_1)^\top:\gH^2\to \cH_0\oplus\cH_1$ such that $\ov{\dom\G}=A^*$, the Green’s identity holds and a certain maximality condition is satisfied (here as before $\cH_0$ is a Hilbert space and $\cH_1$ is a subspace in $\cH_0$). Moreover, by means of the equality $$\begin{gathered}
\label{1.6}
M_+(\l)=\{\{\G_0\{f_\l,\l f_\l\}, \G_1\{f_\l,\l f_\l\}\}: f_\l\in\ker (A^*-\l)
\}, \quad\l\in\bC_+\end{gathered}$$ we associate with a boundary relation $\G$ the Weyl family $M_+(\l)$.

In the paper we study substantially the boundary relations $\BR$ with $\dim\cH_0 <\infty$. We show that in this case $\dom\G=A^*$ and there is a boundary triplet $\Pi_\G=\{\cK_0\oplus\cK_1,G_0, G_1\}$ for $A^*$ with $\cK_j\in\cH_j, \; j\in \{0,1\}$ such that $\G$ can be represented roughly speaking as a 