Language processing is a highly integrative function, intertwining linguistic operations (processing the language code intentionally used for communication) and extra-linguistic processes (e.g., attention monitoring, predictive inference, long-term memory). This synergetic cognitive architecture requires a distributed and specialized neural substrate. Brain systems have mainly been examined at rest. However, task-related functional connectivity provides additional and valuable information about how information is processed when various cognitive states are involved. We gathered thirteen language fMRI tasks in a unique database of one hundred and fifty neurotypical adults (InLang [Interactive networks of Language] database), providing the opportunity to assess language features across a wide range of linguistic processes. Using this database, we applied network theory as a computational tool to model the task-related functional connectome of language (LANG atlas). The organization of this data-driven neurocognitive atlas of language was examined at multiple levels, uncovering its major components (or crucial subnetworks), and its anatomical and functional correlates. In addition, we estimated its reconfiguration as a function of linguistic demand (flexibility) or several factors such as age or gender (variability). We observed that several discrete networks could be specifically shaped to promote key functional features of language: coding-decoding (Net1), control-executive (Net2), abstract-knowledge (Net3), and sensorimotor (Net4) functions. The architecture of these systems and the functional connectivity of the pivotal brain regions varied according to the nature of the linguistic process, gender, or age. By accounting for the multifaceted nature of language and modulating factors, this study can contribute to enriching and refining existing neurocognitive models of language. The LANG atlas can also be considered a reference for comparative or clinical studies involving various patients and conditions.