Observational measurements of the temperature variation of the Cosmic Microwave Background across the celestial sphere made by the Wilkinson Microwave Anisotropy Probe (WMAP) and, more recently Planck, have played a major part in establishing the standard "concordance" cosmological model. However, extensive statistical analysis of these data have also revealed some tantalising anomalies whose interpretation within the standard framework is by no means clear. In this talk, I'll discuss the significance of the evidence for some aspects of this anomalous behaviour, offer some possible theoretical models, and suggest how future measurements may provide firmer conclusions.