CMB anomalies after Planck

Several unexpected features have been observed in the microwave sky at large angular scales, both by WMAP and by Planck. Among those features is a lack of both variance and correlation on the largest angular scales, alignment of the lowest multipole moments with one another and with the motion and geometry of the solar system, a hemispherical power asymmetry or dipolar power modulation, a preference for odd parity modes and an unexpectedly large cold spot in the Southern hemisphere. The individual p-values of the significance of these features are in the per mille to per cent level, when compared to the expectations of the best-fit inflationary ACDM model. Some pairs of those features are demonstrably uncorrelated, increasing their combined statistical significance and indicating a significant detection of CMB features at angular scales larger than a few degrees on top of the standard model. Despite numerous detailed investigations, we still lack a clear understanding of these large-scale features, which seem to imply a violation of statistical isotropy and scale invariance of inflationary perturbations. In this contribution we present a critical analysis of our current understanding and discuss several ideas of how to make further progress.