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Advances in synthetic polymer chemistry have unleashed seemingly unlimited strategies for producing block polymers with arbitrary numbers (n) and types (k) of unique sequences of repeating units. Increasing (k,n) leads to a geometric expansion of possible molecular architectures, beyond conventional ABA-type triblock copolymers (k = 2, n = 3), offering alluring opportunities to generate exquisitely tailored materials with unparalleled control over nanoscale-domain geometry, packing symmetry, and chemical composition. Transforming this potential into targeted structures endowed with useful properties hinges on imaginative molecular designs guided by predictive theory and computer simulation. Here, we review recent developments in the field of block polymers.