We summarise important recent advances in quantum metrology, in connection to experiments in cold gases, trapped cold atoms and photons. First we review simple metrological setups, such as quantum metrology with spin squeezed states, with Greenberger-Horne-Zeilinger states, Dicke states and singlet states. We calculate the highest precision achievable in these schemes. Then, we present the fundamental notions of quantum metrology, such as shot-noise scaling, Heisenberg scaling, the quantum Fisher information and the Cramér-Rao bound. Using these, we demonstrate that entanglement is needed to surpass the shot-noise scaling in very general metrological tasks with a linear interferometer. We discuss some applications of the quantum Fisher information, such as how it can be used to obtain a criterion for a quantum state to be a macroscopic superposition. We show how it is related to the the speed of a quantum evolution, and how it appears in the theory of the quantum Zeno effect. Finally, we explain how uncorrelated noise limits the highest achievable precision in very general metrological tasks.

We present several entanglement criteria in terms of the quantum Fisher information that help to relate various forms of multipartite entanglement to the sensitivity of phase estimation. We show that genuine multipartite entanglement is necessary to reach the maximum sensitivity in some very general metrological tasks using a two-arm linear interferometer. We also show that it is needed to reach the maximum average sensitivity in a certain combination of such metrological tasks.