The Lyapunov spectrum of a map \(g\) is the map \(l(\beta)\) which assigns to any \(\beta\in \mathbb{R}\) the Hausdorff dimension of the \(\beta\)-level set of the Lyapunov exponent of \(g\). In the paper it is shown that if \(g\) is a conformal expanding map (examples of such maps include Markov maps of an interval, rational maps with hyperbolic Julia sets and conformal toral endomorphisms) and the measure of maximal dimension does not have maximal entropy then the Lyapunov spectrum \(l(\beta)\) is real analytic and strictly convex on an interval (i.e., the range of the Lyapunov exponent contains an interval) and the level sets corresponding to this interval are dense in the phase space. Similar results are proved for most Axiom-A surface diffeomorphisms. Also some rigidity results are obtained for rational maps and geodesic flows.