This chapter will introduce the reader to the study of third (L3) or further (Ln) acquisition. The circumstances surrounding the acquisition of language beyond the second conspire to make this a unique context, distinct from L2 acquisition. The L3 learner has at least partial knowledge of more than one grammar, previous experience with nonnative language learning and, in most cases, at least some metalinguistic knowledge. All these factors significantly condition both the starting point and the trajectory of L3/Ln language acquisition in ways that cannot be predicted from our understanding of L1 and L2 acquisition and reflect qualitative as well as quantitative changes in going from two to three or more languages. In this chapter, the reader will find an introduction to the basic concepts and themes of L3/Ln acquisition research, an overview of L3/Ln theories from a cognitive perspective – modeling the acquisition of both morphosyntax and phonology – an in-depth discussion of the role of previously acquired languages, and a brief introduction to some of the most recent work on L3 processing, including studies using electroencephalography (EEG) and semiartificial languages to address long-standing questions in the field while overcoming some of the most common methodological and logistic issues facing the L3/Ln researcher.