The physics of many-body systems strongly depends on their dimensionality. For example, in a two-dimensional world, most standard phase transitions towards an ordered state of matter would not occur because of the increased role of fluctuations. However, unconventional "topological" transitions can still take place in Flatland, and novel phases of matter also characterized by a non-trivial topology - e.g. Quantum Hall states - can emerge. The 2016 Physics Nobel prize was awarded to Kosterlitz, Thouless and Haldane for their pioneering work in this field.

In this set of lectures I will describe recent experimental investigations of Flatland physics performed with cold atomic gases. In particular I will present the realization of uniform atomic gases confined in ``square box" potentials, which has opened a number of perspectives in the physics of quantum fluids. These uniform systems are particularly relevant for the study of critical phenomena where one expects a divergence of correlation lengths. I will also review the recent progress in the formation of topological matter with cold atom setups, based on the interplay between light and matter and/or a periodic drive of the system.