General aspects of spectral theory of elliptic operators on non-compact manifolds are studied. Methods of proving the coincidence of minimal and maximal operators are descibed and a review of the known results are given. Exponential weight estimates for the decay of the Green function on manifolds of bounded geometry are proved. Applications of these estimates to Schnol type theorems are given (these theorem give conditions of growth to be imposed on a nontrivial generazed eigenfunction to garantee that the corresponding eigenvalue is in the spectrum). This is done in particular on manifold of bounded geometry with the exponential growth of the volume of the balls. Estimates of growth of generalized eigenfunctions for almost all points in the spectrum (with respect to the spectral measure) are given.