Research on the brain basis of speech and language faces theoretical and empirical challenges. Most current research, dominated by imaging, deficit-lesion, and electrophysiological techniques, seeks to identify regions that underpin aspects of language processing such as phonology, syntax, or semantics. The emphasis lies on localization and spatial characterization of function. The first part of the paper deals with a practical challenge that arises in the context of such a research programme. This maps problem concerns the extent to which spatial information and localization can satisfy the explanatory needs for perception and cognition. Several areas of investigation exemplify how the neural basis of speech and language is discussed in those terms (regions, streams, hemispheres, networks). The second part of the paper turns to a more troublesome challenge, namely how to formulate the formal links between neurobiology and cognition. This principled problem thus addresses the relation between the primitives of cognition (here speech, language) and neurobiology. Dealing with this mapping problem invites the development of linking hypotheses between the domains. The cognitive sciences provide granular, theoretically motivated claims about the structure of various domains (the "cognome"); neurobiology, similarly, provides a list of the available neural structures. However, explanatory connections will require crafting of computationally explicit linking hypotheses at the right level of abstraction. For both the practical maps problem and the principled mapping problem, developmental approaches and evidence can play a central role in the resolution.