In the presented paper, the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator is developed. The analysis of this paper is the development of ideas coming from the application of the classical Fourier series techniques in the analysis of pseudo-differential operators on the torus to a more general setting without assuming that the problem has symmetries. Nevertheless, the authors attempt to still mimic the harmonic analysis constructions but in the new “nonharmonic” setting. For this purpose, they establish elements of non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. At the end of the paper applications of the developed analysis to obtain a priori estimates for solutions of boundary value problems that are elliptic within the constructed calculus are given.