It is well-known that many quantum observables cannot be measured simultaneously, a prominent example being the position and momentum observables of a quantum mechanical particle. This property is referred to as incompatibility of measurements, and in this talk I will focus on the incompatibility of finite-outcome measurements in finite dimensions. Beyond the binary compatible / incompatible characterisation, one can quantify to what extent a pair of measurements is incompatible. One way to do so is to quantify the robustness of incompatibility with respect to some noise. There are many such measures known in the literature, with the main difference between them being the specific noise model. In this talk I will introduce the basic framework for robustness-based incompatibility measures, and discuss some natural properties that they should ideally satisfy. Then I will discuss five of the most widely used measures, analyse their properties, and provide generic and measurement-specific bounds on them. For one measure, we show that the most incompatible pair of measurements corresponds to mutually unbiased bases. For other measures this question remains open, but we show that the most incompatible measurements depend on the chosen incompatibility measure in general.