Let M be a complete Riemannian manifold of dimension n. Suppose that the sectional curvatures K of M are negative and pinched by \(-1\leq K\leq - 1+\epsilon\), \(0\leq\epsilon <1\). The paper concerns the spectrum of the Laplacian \(\Delta_ p\) acting on square integrable p-forms on M. One assumes that M is either simply connected or of finite volume. In the simply connected case, a lower bound on the spectrum of \(\Delta_ p\) is obtained. In particular, all square integrable harmonic p-forms must vanish for \(p=n/2\), \((n\pm 1)/2\), and if \(\epsilon <1-(n-2)^ 2/(n-1)\). This establishes, in part, a conjecture of J. Dodziuk and I. M. Singer. If M has finite volume, a lower bound is given for the essential spectrum of \(\Delta_ p\). The space of square integrable harmonic p-forms is finite dimensional for \(p\neq (n\pm 1)/2\).