We present the approach to the study of language acquisition and processing developed in LAPAL (Laboratório de Psicolinguística e Aquisição da Linguagem – PUC-Rio) in the last decades, which incorporates a minimalist conception of language in a procedural model of grammar identification, and in the online computation of sentences. We refer to experimental results regarding infants’ sensibility to elements of functional categories, the early acquisition of gender and number, the early parsing of recursive structures, the role of recursion in cognitive development, and the processing cost of recursive structures and long-distance dependencies. Methodological issues concerning the assessment of children’s ability to process costly structures are considered, which contribute to the identification of the nature of children’s difficulties in comprehension tasks. Particular attention is paid to the application of this approach to the tracking of DLD (Developmental Language Disorder).