Embryonic morphogenesis occurs along three orthogonal axes. While the patterning of the anterior-posterior and dorsal-ventral axes has been increasingly well-characterized, the left-right (LR) axis has only relatively recently begun to be understood at the molecular level. The mechanisms that ensure invariant LR asymmetry of the heart, viscera, and brain involve fundamental aspects of cell biology, biophysics, and evolutionary biology, and are important not only for basic science but also for the biomedicine of a wide range of birth defects and human genetic syndromes. The LR axis links biomolecular chirality to embryonic development and ultimately to behavior and cognition, revealing feedback loops and conserved functional modules occurring as widely as plants and mammals. This review focuses on the unique and fascinating physiological aspects of LR patterning in a number of vertebrate and invertebrate species, discusses several profound mechanistic analogies between biological regulation in diverse systems (specifically proposing a nonciliary parallel between kidney cells and the LR axis based on subcellular regulation of ion transporter targeting), highlights the possible importance of early, highly-conserved intracellular events that are magnified to embryo-wide scales, and lays out the most important open questions about the function, evolutionary origin, and conservation of mechanisms underlying embryonic asymmetry.