The theory of Computational Singular Perturbation (CSP) has been widely used in the chemical kinetics and combustion community to understand complex chemistry, reduce chemical kinetic models and perform computational diagnostics in reacting flow simulations. Useful as it is, the application of CSP is nevertheless subject to computationally expensive eigen-decomposition of the chemical Jacobian matrix, especially for mechanisms involving a large number of species. In 2016-2017, a new CSP approach was developed by Lam, taking advantage of species lifetime and linearity in the governing equation of the fastest species. The corresponding quasi-state value of the current fastest species across the thin transition layer could be accurately predicted at each time step, during which period the concentration of the slower species is considered as constant. As a result, tedious explicit integration to resolve the apparent exponential transition layer, as those utilized in conventional implicit chemistry solvers (e.g., CVODE), is avoided and the integration time step could be largely extended. This method, referred to as linear CSP (LCSP), has been demonstrated to be useful in deriving reduced models and to numerically integrate stiff ODE systems. In 2018, an idea of nonlinear CSP (NCSP) following similar approach was further proposed by Lam, which abandons the enabling assumptions of the linearized governing equation for the fastest species and provides useful insights into the numerical integration of stiff chemical ODE systems. The development of CSP without expensive eigen-decomposition and computational evidence supporting its implementation are discussed and disseminated as a promising direction of advanced CSP theory. For general kinetic problems with complex couplings, a method based on sparse adaptive hybrid integration is introduced to demonstrate the redundancy issue and general stiffness in the fast subspace. (C) 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.