The purpose of this article is to establish a functional Large Deviations Principle (LDP) for L-statistics under conditions on the extremes. The method is based on Sanov’s theorem and the usual tools of the theory of large deviations. We first prove a LDP under a quite strong extremes condition. We provide the full treatment of the case of the uniform distribution and an example in which the rate function can be calculated very precisely. Afterwards, we obtain a LDP under weaker extremes conditions. The case of the exponential distribution, which does not match the former integrability conditions, is treated owing to another method: we give a functional LDP based on the Gärtner-Ellis theorem. We extend our study to normalized L-statistics under strong extremes conditions.