  As we enter the era of quantum technologies, quantum estimation theory
provides an operationally motivating framework for determining high precision
devices in modern technological applications. The aim of any estimation process
is to extract information from an unknown parameter embedded in a physical
system such as the estimation converges to the true value of the parameter.
According to the Cram\'er-Rao inequality in mathematical statistics, the Fisher
information in the case of single-parameter estimation procedures, and the
Fisher information matrix in the case of multi-parameter estimation, are the
key quantities representing the ultimate precision of the parameters specifying
a given statistical model. In quantum estimation strategies, it is usually
difficult to derive the analytical expressions of such quantities in a given
quantum state. This review provides comprehensive techniques on the analytical
calculation of the quantum Fisher information as well as the quantum Fisher
information matrix in various scenarios and via several methods. Furthermore,
it provides a mathematical transition from classical to quantum estimation
theory applied to many freedom quantum systems. To clarify these results, we
examine these developments using some examples. Other challenges, including
their links to quantum correlations and saturating the quantum Cram\'er-Rao
bound, are also addressed.
