Learning the language of biological sequences is an appealing challenge for the grammatical inference research field. While some first successes have already been recorded, such as the inference of profile hidden Markov models or stochastic context-free grammars which are now part of the classical bioinformatics toolbox, it is still a source of open and nice inspirational problems for grammatical inference, enabling us to confront our ideas to real fundamental applications. As an introduction to this field, we survey here the main ideas and concepts behind the approaches developed in pattern/motif discovery and grammatical inference to characterize successfully the biological sequences with their specificities.