Our study is based on operators approach. First, positive integer s order fractional integral is defined by iterating s-times one order integral using Euler’s gamma functions properties. Then, the definition is extended to any value of s (positive , negative, fractional, transcendental, real, complex numbers) using the extension of Euler’s gamma and beta functions. Many properties (limits, linearity, semi-group) are given. The most important and useful property is the semi-group one Then , the definition of fractional derivative is derived from this property. There are two types of fractional derivatives, left handed and right handed ones. The left handed fractional derivative applied to a constant function gives a non null result whereas the right handed one leads to a null result. The latter one is then better than the first one. In a previous work, we have studied the case of fractional operators applied to the set of causal functions. In the present one, we look for the set of anti-causal functions. We introduce anti-integral and anti-derivatives operators. Though properties obtained for the cases of causal and anti- causal functions are similar, there are some differences. We obtain formulae given by many authors (Liouville, Riemann, Caputo, Liouville-Caputo) as particular cases of ours.