We introduce and study a general pseudodifferential calculus for boundary value problems on a class of non-compact manifolds with boundary (so-called Lie manifolds with boundary). This is accomplished by constructing a suitable generalization of the Boutet de Monvel calculus for boundary value problems. The data consists of a compact manifold with corners $M$ that is endowed with a Lie structure of vector fields $\mathcal{V}$, a so-called Lie manifold. The manifold $M$ is split into two equal parts $X_{+}$ and $X_{-}$ which intersect in an embedded hypersurface $Y \subset X_{\pm}$. Our goal is to describe a transmission Boutet de Monvel calculus for boundary value problems compatible with the structure of Lie manifolds. Starting with the example of $b$-vector fields, we show that there are two groupoids integrating the Lie structures on $M$ and on $Y$, respectively. These two groupoids form a bibundle (or a groupoid correspondence) and, under some mild assumptions, these groupoids are Morita equivalent. With the help of the bibundle structure and canonically defined manifolds with corners, which are blow-ups in particular cases, we define a class of Boutet de Monvel type operators. We then define the representation homomorphism for these operators and show closedness under composition with the help of a representation theorem. Finally, we consider appropriate Fredholm conditions and construct the parametrices for elliptic operators in the calculus.

41 pages, revised version