Open and marginally stable quantum systems are in the center of current interest in many branches of mesoscopic physics, from nuclei to future quantum computers. A powerful method of their theoretical description reformulates the exact quantum mechanics in the language of the effective non-Hermitian Hamiltonian. The whole dynamics is projected into that of intrinsic system with unstable quantum states. The techniques are explained, and examples are given for modern nuclear shell model with continuum and for quantum signal transmission through a mesoscopic system. An analogy to super-radiance in quantum optics plays an important role.