The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes, especially stochastic integrals and differential equations. In this paper, general central limit theorems and functional central limit theorems are obtained for martingale like random variables under the sub-linear expectation. As applications, the Lindeberg central limit theorem and functional central limit theorem are obtained for independent but not necessarily identically distributed random variables, and a new proof of the Lévy characterization of a G-Brownian motion without using stochastic calculus is given. For proving the results, we have also established Rosenthal's inequality and the exceptional inequality for the martingale like random variables.