Abstract <p>The article studies the basic properties of systems of eigen- and associated functions of ordinary differential operators of high order with integral conditions. Conditions for the unconditional basis property of these systems in the space of vector–functions square-summable on a finite interval of the real axis are established. Integral representations of solutions of spectral problems for the adjoint operator are obtained, estimates of these solutions in various metrics are established. Conditions for the validity of Bessel inequalities for both the direct and adjoint operators are found.</p>