Ancient Mesopotamians lived their lives in a complex urban environment in a universe that was governed by rules and regulations that were believed to have been set in place by the gods. Within this framework, Mesopotamian scholars produced cuneiform texts which at times hint at notions of infinity that may be compared to popular modern ideas on this subject: infinite time, infinite space, infinite stars, and even a sense of theoretical mathematical infinity. This lecture presents the evidence for an Ancient Near Eastern sense of the infinite, and open a dialogue between modern and ancient understandings of the term.