We view the classical Lindeberg principle in a homogeneous Markov process setting to establish a probability approximation framework by the associated It\^{o}'s formula and Markov operator. As applications, we study the error bounds of the following three approximations: multivariate normal approximation, approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by multiplicative Brownian motion, and Euler-Maruyama (EM) discretization for SDEs driven by \alpha-stable process. Furthermore, we extend the above approximation framework to the inhomogeneous case and consider the variable-step E-M approximations of regime-switching jump diffusion processes. This talk is based on the joint works with Q . M. Shao, Z. G. Su, C. Deng and L. Xu .