The Groenewold-Moyal plane and its quantum physics

10 pages, no figures.-- PACS nrs.: 03.70.+k, 02.30.Nw, 11.10.Nx, 04.60.Bc.-- ArXiv pre-print available at: http://arxiv.org/abs/0908.3888v1.pdfThis work is based on the talk given by APB at the XXV Max Born Symposium on Planck Scale Physics, Wroclaw, 29 June-3 July 2009.Quantum theories constructed on the noncommutative spacetime called the Groenewold-Moyal(GM) plane exhibit many interesting properties such as causality violation, Lorentz and CPT non-invariance and twisted statistics. Such violations lead to many striking features that may be tested experimentally. Thus these theories predict Pauli-forbidden transitions due to twisted statistics, anisotropies and acausal effects in the cosmic microwave background radiation in correlations of observables and Lorentz and CPT violations in scattering amplitudes. Such features of quantum physics on the GM plane are surveyed in this review.APB thanks Alberto Ibort and the Universidad Carlos III de Madrid for their kind hospitality and support. This work was partially supported by the US Department of Energy under grant number DEFG02-85ER40231.Publicad