Perception and cognition are teeming with signals and concepts that interact in a multiplicative way, and it is largely this pattern of combination that generates the awesome variability and complexity of our physical and mental worlds. Human brains are prodigious in contending with such complexity, in part due to their ability to factor concepts into more fundamental parts. This talk will cover our attempts to confront the challenge (and promise) of multiplicative representations, and their attendant factorization problems, in the brain. We have focused on a paradigm for modeling cognition that defines an algebra over high-dimensional vectors and presents a compelling factorization problem. Our solution to this problem, a recurrent neural network architecture we call Resonator Networks, has several interesting properties that make it uniquely effective on this problem and may provide some principles for designing a new class of neural network models. We show some applications of multiplicative distributed codes for representing visual scenes and suggest how such representations may be a useful tool for unifying symbolic and connectionist theories of intelligence.