Visual object recognition -- the behavioral ability to rapidly and accurately categorize many visually encountered objects -- is core to primate cognition. This behavioral capability is algorithmically impressive because of the myriad identity-preserving viewpoints and scenes that dramatically change the visual image produced by the same object. Until recently, the brain mechanisms that support that capability were deeply mysterious. However, over the last decade, this scientific mystery has been illuminated by the discovery and development of brain-inspired, image-computable, artificial neural network (ANN) systems that rival primates in this behavioral feat. Apart from fundamentally changing the landscape of artificial intelligence (AI), modified versions of these ANN systems are the current leading scientific hypotheses of an integrated set of mechanisms in the primate ventral visual stream that support object recognition. What separates brain-mapped versions of these systems from prior conceptual models is that they are Sensory-computable, Mechanistic, Anatomically Referenced, and Testable (SMART). Here, we review and provide perspective on the brain mechanisms that the currently leading SMART models address. We review the empirical brain and behavioral alignment successes and failures of those current models. Given ongoing advances in neurobehavioral measurements and AI, we discuss the next frontiers for even more accurate mechanistic understanding. And we outline the likely applications of that SMART-model-based understanding.