These notes are a translation into Englsh of the preprint "Théorèmes limites pour les suites de variables aléatoires faiblement dépendantes", Prépublication 97-81 de l'Université de Paris-Sud" which was published at the end of the year 1997. This course aims to treat the theory of summation of weakly dependent variables. The first eight chapters give extensions of the classical results for sums of independent random variables to strongly mixing or absolutely regular processes. Chapter 9 is devoted to applications to Markov chains. The potential reader is any researcher who is interested in sharp results for weakly dependent sequences.