  Quantum metrology holds the promise of an early practical application of
quantum technologies, in which measurements of physical quantities can be made
with much greater precision than what is achievable with classical
technologies. In this review, we collect some of the key theoretical results in
quantum parameter estimation by presenting the theory for the quantum
estimation of a single parameter, multiple parameters, and optical estimation
using Gaussian states. We give an overview of results in areas of current
research interest, such as Bayesian quantum estimation, noisy quantum
metrology, and distributed quantum sensing. We address the question how minimum
measurement errors can be achieved using entanglement as well as more general
quantum states. This review is presented from a geometric perspective. This has
the advantage that it unifies a wide variety of estimation procedures and
strategies, thus providing a more intuitive big picture of quantum parameter
estimation.
