Sigmoid functions, named after their distinctive S-shaped curve, are commonly used to model various phenomena in fields such as health (e.g., COVID-19), mathematics, artificial intelligence, deep learning, neural networks, economics, econometrics, computer science, biology, and other disciplines. Sigmoid functions have not been thoroughly studied until now, nor is the theory of sigmoid functions well developed. This study is very likely to be the first attempt to conduct systematic research towards the general theory of sigmoid functions. The key result of this study is an extension to the traditional definition of sigmoid functions. While sigmoid functions are typically defined as monotonic and increasing, we introduce non-monotonic sigmoid functions. For the first time, we introduce three (3) kinds of sigmoid functions and define their properties. Moreover, the key result is also thirty (30) construction methods for unitary sigmoid functions that are presented in the taxonomy. We also introduce the method to construct filtered sigmoid functions. They possess the unique property of filtering a standard sigmoid function into a smoother, more gradual curve. Additionally, the method for constructing unitary sigmoid functions uses recursive discretization of the sigmoid differential equation. Stochastic sigmoid function is presented too. We intend to publish around thirty additional studies presenting more sigmoid functions related to the methods.