In the last decade, artificial intelligence (AI) models inspired by the brain have made unprecedented progress in performing real-world perceptual tasks like object classification and speech recognition. Recently, researchers of natural intelligence have begun using those AI models to explore how the brain performs such tasks. These developments suggest that future progress will benefit from increased interaction between disciplines. Here we introduce the Algonauts Project as a structured and quantitative communication channel for interdisciplinary interaction between natural and artificial intelligence researchers. The projects core is an open challenge with a quantitative benchmark whose goal is to account for brain data through computational models. This project has the potential to provide better models of natural intelligence and to gather findings that advance AI. The 2019 Algonauts Project focuses on benchmarking computational models predicting human brain activity when people look at pictures of objects. The 2019 edition of the Algonauts Project is available online: http://algonauts.csail.mit.edu/.