We demonstrate that the key components of cognitive architectures - declarative and procedural memory - and their keycapabilities - learning, memory retrieval, judgement, and decision-making - can be implemented as algebraic operationson vectors in a high-dimensional space. Modern machine learning techniques have an impressive ability to process datato find patterns, but typically do not model high-level cognition. Traditional, symbolic cognitive architectures can capturethe complexities of high-level cognition, but have limited ability to detect patterns or learn. Vector-symbolic architec-tures, where symbols are represented as vectors, bridge the gap between these two approaches. Our vector-space modelaccounts for primacy and recency effects in free recall, the fan effect in recognition, human probability judgements, andhuman performance on an iterated decision task. Our model provides a flexible, scalable alternative to symbolic cognitivearchitectures at a level of description that bridges symbolic, quantum, and neural models of cognition.