This paper introduces a model-based compressive sensing (CS) theory that extends the conventional CS framework by leveraging more realistic signal models that go beyond simple sparsity and compressibility. The authors propose a new class of structured compressible signals and a sufficient condition for robust recovery, called the restricted amplification property (RAMP), which is the counterpart to the restricted isometry property (RIP) in conventional CS. The paper demonstrates how to integrate structured sparsity models into two state-of-the-art CS recovery algorithms, CoSaMP and iterative hard thresholding (IHT), and proves that these algorithms can robustly recover signals from significantly fewer measurements. Two specific examples, wavelet tree and block sparsity, are used to validate the theory and algorithms through numerical simulations, showing significant performance gains over standard CS methods. The paper also discusses the robustness of model-based CS recovery to model mismatch and provides computational complexity analysis.This paper introduces a model-based compressive sensing (CS) theory that extends the conventional CS framework by leveraging more realistic signal models that go beyond simple sparsity and compressibility. The authors propose a new class of structured compressible signals and a sufficient condition for robust recovery, called the restricted amplification property (RAMP), which is the counterpart to the restricted isometry property (RIP) in conventional CS. The paper demonstrates how to integrate structured sparsity models into two state-of-the-art CS recovery algorithms, CoSaMP and iterative hard thresholding (IHT), and proves that these algorithms can robustly recover signals from significantly fewer measurements. Two specific examples, wavelet tree and block sparsity, are used to validate the theory and algorithms through numerical simulations, showing significant performance gains over standard CS methods. The paper also discusses the robustness of model-based CS recovery to model mismatch and provides computational complexity analysis.