Artificial Neural Networks (ANNs) inspired by biology are beginning to be widely used to model behavioral and neural data, an approach we call neuroconnectionism. ANNs have been lauded as the current best models of information processing in the brain, but also criticized for failing to account for basic cognitive functions. We propose that arguing about the successes and failures of a restricted set of current ANNs is the wrong approach to assess the promise of neuroconnectionism. Instead, we take inspiration from the philosophy of science, and in particular from Lakatos, who showed that the core of scientific research programmes is often not directly falsifiable, but should be assessed by its capacity to generate novel insights. Following this view, we present neuroconnectionism as a cohesive large-scale research programme centered around ANNs as a computational language for expressing falsifiable theories about brain computation. We describe the core of the programme, the underlying computational framework and its tools for testing specific neuroscientific hypotheses. Taking a longitudinal view, we review past and present neuroconnectionist projects and their responses to challenges, and argue that the research programme is highly progressive, generating new and otherwise unreachable insights into the workings of the brain.