Because the Lie Theory solely applies to linear systems, in 1978 Santilli proposed the isotopic lifting of Lie’s theory for nonlinear systems, today known as the Lie-Santilli isotheory, via the reconstruction of linearity on the isotopic lifting of spaces and fields. In order to identify the proper mathematical background of the Lie-Santilli isotheory, Kadeisvili introduced in 1992 the notion of isocontinuity; Tsagas and Sourlas proposed in 1995 a for of isotopology defined over conventional fields; Santilli extended it in 1996 its formulation on isofields; and the authors conducted in 2003 a systematic study of the new isotopology. In this paper we outline the foundation of the new isotopology and present various advances
