Abstract A transition edge sensor (TES) is an X-ray microcalorimeter based on a thin film superconductor voltage-biased within its resistive transition. This superconductor is usually modelled as a variable resistor, completely described by a single value of the logarithmic temperature coefficient, α. Recent measurements of excess noise (and poorer than expected X-ray energy resolution) demand, however, a more detailed physical model of the superconducting-to-normal transition. In this paper, we show, using data from Ir films and from Ti- , W- , Ir- , Mo/Cu- and Ti/Au-based devices, that the TES may be plausibly regarded as a two-dimensional superconductor exhibiting a Kosterlitz–Thouless–Berezinsky phase transition due to vortex dynamics. Adopting this model leads to : (i) expressions for the variation of both TES resistance and the α-parameter with temperature (or bias voltage) within the transition and (ii) a physical description of the excess noise source and its scaling laws, together with strategies for its reduction.