  Solutions of Stochastic Differential Equations can have three types of
explosive behaviors: almost-sure non-explosive, explosion with positive
probability, and almost sure explosion. In this paper, we will provide a survey
of Khasminskii's Lyapunov method for classifying explosive behaviors of
solutions of stochastic differential equations. We will embark our expedition
by examining the renowned Feller's test for explosion and observing its
shortfalls. Afterwards, we will present Khasminskii's Lyapunov method for
almost-sure non-explosion, explosion with positive probability, and almost-sure
explosion. Ample examples will be provided to illuminate the power of
Khasminskii's Lyapunov methods. Furthermore, quick layovers will be made to
extend Khasminskii's Lyapunov method for almost-sure non-explosion and
explosion with positive probability for jump processes with constant Poisson
intensities.
