Turbulent flows are chaotic and in three dimensions tend to destroy any order present in the flow. This behaviour changes drastically in two dimensions, where nonlinear processes lead to accumulation of energy at the largest scale through an inverse cascade, thus spontaneously ordering the flow into large-scale eddies. Superfluid helium is uniquely suited for the study of two-dimensional and quasi-two-dimensional turbulence, since the inviscid liquid easily flows into narrow channels and the interaction of quantized vorticity with thermal excitations (mutual friction) allows varying the dissipation suffered by the flow. Multiple types of critical phenomena in transition to turbulence are observed, with both flow velocity and temperature as control variables. The observed behaviour can be qualitatively explained through population dynamics of positive and negative discrete point vortices.