INTRODUCTION This article is directed toward those interested in the comparative study of how complex societies develop~urban and historical geographers, urban historians and sociologists, social and economic historians, or anthropologists in or out of Mesoamerican archaeology. It concentrates on aspects of complexity that can be most readily compared cross-culturally. Specifically the article describes formal characteristics of settlement hierarchies in the Valley of Oaxaca, in southern Mexico, from their beginnings at about 1500 Be until the Spanish Conquest 3000 years later. I omit the archaeological and analytical details and the intricate variations since a book-length review of the prehistory of the Valley and the adjacent Mixteca Alta and a technical monograph on the archaeological settlement patterns have recently been published (32, 45). Even without their details, the new archaeological studies of regions make it apparent that social science theories about state and urban evolution need to be upgraded to account for unexpected variation and unforeseen regularities. By complexity I mean the multiplicity of different parts in a social system (12; cf 52 for a formal approach to cultural complexity). To monitor complexity one must be able to identify parts, specify that they are different, be sure that one has all the parts, count them, and ascertain that they form a system. Meeting these five data requirements is not easy, with either historical or archaeological data. The realm of the social is complex in this sense with its