Quantum Mechanics puts a limit on how small the degradation of information passing through an amplifier can be. It is known theoretically that the minimum noise added by an amplifier to the signal amounts, in phase preserving mode, to at least half a photon at the signal frequency. Is it possible to construct an amplifier working at microwave frequencies that would reach this quantum limit and still be practical in terms of gain, bandwidth and dynamical range? We will present recent work by our group aiming at answering this question, which is of crucial importance for the readout of solid state qubits, and more generally, for measurement of very weak electromagnetic signals in various areas of science.