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As we enter the era of quantum technologies, quantum estimation theoryprovides an operationally motivating framework for determining high precisiondevices in modern technological applications. The aim of any estimation processis to extract information from an unknown parameter embedded in a physicalsystem such as the estimation converges to the true value of the parameter.According to the Cram\'er-Rao inequality in mathematical statistics, the Fisherinformation in the case of single-parameter estimation procedures, and theFisher information matrix in the case of multi-parameter estimation, are thekey quantities representing the ultimate precision of the parameters specifyinga given statistical model. In quantum estimation strategies, it is usuallydifficult to derive the analytical expressions of such quantities in a givenquantum state. This review provides comprehensive techniques on the analyticalcalculation of the quantum Fisher information as well as the quantum Fisherinformation matrix in various scenarios and via several methods. Furthermore,it provides a mathematical transition from classical to quantum estimationtheory applied to many freedom quantum systems. To clarify these results, weexamine these developments using some examples. Other challenges, includingtheir links to quantum correlations and saturating the quantum Cram\'er-Raobound, are also addressed.

Analytical techniques in single and multi-parameter quantum estimation theory: a focused review