We study the spectrum of complete non-compact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. As applications, we construct metrics with an arbitrarily large finite number of gaps in its essential spectrum on non-compact covering of a compact manifold and complete non-compact manifold with bounded curvature and positive injectivity radius.This is a joint work with Richard Schoen.