In the last decade, sequential Monte-Carlo methods (SMC) emerged as a key tool in computational statistics (see for instance [1], [2], [3]). These algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighted population of particles, which are generated recursively. Despite many theoretical advances (see for instance [4], [5], [6], [7]), the large-sample theory of these approximations remains a question of central interest. In this paper, we establish a law of large numbers and a central limit theorem as the number of particles gets large. We introduce the concepts of weighted sample consistency and asymptotic normality, and derive conditions under which the transformations of the weighted sample used in the SMC algorithm preserve these properties. To illustrate our findings, we analyze SMC algorithms to approximate the filtering distribution in state- space models. We show how our techniques allow to relax restrictive technical conditions used in previously reported works and provide grounds to analyze more sophisticated sequential sampling strategies, including branching, resampling at randomly selected times, etc..