Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for acquisition of sparse or compressible signals that can be well approximated by just K << N elements from an N-dimensional basis. Instead of taking periodic samples, we measure inner products with M < N random vectors and then recover the signal via a sparsity-seeking optimization or greedy algorithm. The standard CS theory dictates that robust signal recovery is possible from M = O(K log(N/K)) measurements. The goal of this paper is to demonstrate that it is possible to substantially decrease M without sacrificing robustness by leveraging more realistic signal models that go beyond simple sparsity and compressibility by including dependencies between values and locations of the signal coefficients. We introduce a model based CS theory that parallels the conventional theory and provides concrete guidelines on how to create model-based recovery algorithms with provable performance guarantees. A highlight is the introduction of a new class of model-compressible signals along with a new sufficient condition for robust model compressible signal recovery that we dub the restricted amplification property (RAmP). The RAmP is the natural counterpart to the restricted isometry property (RIP) of conventional CS. To take practical advantage of the new theory, we integrate two relevant signal models — wavelet trees and block sparsity — into two state-of-the-art CS recovery algorithms and prove that they offer robust recovery from just M = O(K) measurements. Extensive numerical simulations demonstrate the validity and applicability of our new theory and algorithms.