Research in the brain and cognitive sciences attempts to uncover the neural mechanisms underlying intelligent behavior in domains such as vision. Due to the complexities of brain processing, studies necessarily had to start with a narrow scope of experimental investigation and computational modeling. I argue that it is time for our field to take the next step: build system models that capture a range of visual intelligence behaviors along with the underlying neural mechanisms. To make progress on system models, we propose integrative benchmarking – integrating experimental results from many laboratories into suites of benchmarks that guide and constrain those models at multiple stages and scales. We show-case this approach by developing Brain-Score benchmark suites for neural (spike rates) and behavioral experiments in the primate visual ventral stream. By systematically evaluating a wide variety of model candidates, we not only identify models beginning to match a range of brain data (~50% explained variance), but also discover that models’ brain scores are predicted by their object categorization performance (up to 70% ImageNet accuracy). Using the integrative benchmarks, we develop improved state-of-the-art system models that more closely match shallow recurrent neuroanatomy and early visual processing to predict primate temporal processing and become more robust, and require fewer supervised synaptic updates. Taken together, these integrative benchmarks and system models are first steps to modeling the complexities of brain processing in an entire domain of intelligence.