Under the classical framework of probability theory in which the expectation is linear, the Lindeberg's central limit theorem is the fundamental tool for studying the convergence of stochastic processes, especially stochastic integrals and differential equations. Non-linear expectations are useful frameworks for studying uncertainties in statistics, measures of risk, superhedging in finance and non-linear stochastic calculus. In this talk we consider a kind of non-linear expectations, called the sub-linear expectations which are introduced by Peng Shige (2006, 2008). We will present Lindeberg's central limit theorems and functional central limit theorems for independent random variables as well as martingale like random variables under the sub-linear expectations. For establishing the results, Rosenthal's inequality and the exceptional inequality for the martingale like random variables are also given