The historical developments, parametrization, and numerous applications of algebraic numerical differentiation methods initiated by Fliess, Mboup, Join, and Sira-Ramírez are presented. These numerical non-asymptotic approximation approaches for higher-order derivatives of noisy signals are suited for real-time embedded systems. The different derivation approaches are reviewed according to their historical developments. Links to numerous established numerical differentiation algorithms like the Savitzky-Golay filters are discussed. Recent tuning approaches that are reviewed facilitate a parametrization for a good estimation accuracy without relying on trial-and-error approaches, which is essential for industrial applications. The use of these guidelines is demonstrated using two concrete examples: Identification of system parameters from noisy measurements and numerical inversion of the dynamics of analogue anti-aliasing filters. Moreover, an extensive literature survey with various applications of these methods by the control community in all engineering areas is presented. The problems solved include, but are not limited to, parameter estimation, state reconstruction, feedback control, fault diagnosis, anomaly detection, fault-tolerant control, and model-free control. A MATLAB and Python toolbox implementing all necessary functions for the design, analysis, and discretization of the filters is made available, for which a link is provided. Its use is demonstrated in a short appendix.