Rapid advance of quantum technologies demands novel mathematical tools for engineering complex quantum systems. Characterization of the structural and dynamical properties of large-scale quantum devices, e.g., a quantum computer with 100 qubits, is among the current challenges. The major obstacle is the size of the Hilbert space and therefore the required experimental and computational resources that grow exponentially with the number of the system components. Recently, compressed sensing method has been applied for efficient characterization of quantum systems. Originally developed in classical signal processing, compressed sensing is a method to compress high-dimensional signals with a small number of measurements assuming that the signals live on a low-dimensional manifold, and then to reliably reconstruct them. In this presentation, I talk about the compressed sensing theory for quantum inversion problems, its first experimental realization, and the new problems motivated by quantum applications. [1] A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M.P. Almeida, A. Fedrizzi and A. G. White, ”Efficient measurement of quantum dynamics via compressive sensing”, Phys. Rev. Lett 106, 100401 (2011). [2] A. Shabani, M.Mohseni, S. Lloyd, R. L. Kosut and H. Rabitz, ”Estimation of many-body quantum Hamiltonians via compressive sensing”, Phys. Rev. A 84, 012107 (2011).