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The subject of this book is extension of some families of sigmoid functions (log-logistic, hyper-log-logistic, quadratic transmuted log-logistic, exponentiated Weibull-log-logistic, Weibull-R family with baseline Pareto and Lomax cumulative sigmoids, G-Zubair-family with baseline Burr XII cumulative sigmoid, recurrence generated families of sigmoid functions) with applications to growth theory. We also study the one-sided Hausdorff approximation of the generalized cut function by sigmoidal general n-stage growth model. The estimates of the value of the best Hausdorff approximation can be used in practice as one possible additional criterions in ''saturation'' and ''lag-time'' study. Some applications in the field of biochemical processes and debugging theory are also explored. Numerical examples are presented using CAS MATHEMATICA.