I will summarise recent theoretical work on nonlinear response in systems with gapped interacting quasiparticles. I will introduce a quite generally applicable formulation of response of such systems in terms of semiclassical spacetime trajectories. I will then move to two dimensions, using the space-time approach, and outline a proposal to use pump-probe spectroscopy as a sharp diagnostic of fractional statistics in two-dimensionally topologically ordered phases. This promises a route to a positive diagnostic of gapped quantum spin liquids even without momentum-resolved measurements. Along the way, I will comment on a surprising byproduct: the prediction of anomalous thermal broadening of two-point response functions in systems hosting mobile anyon as low-energy gapped quasiparticles. [Based on Phys. Rev. Lett. 131, 256505 (2023); Phys. Rev. Lett. 132, 066702 (2024); Phys. Rev. B 109, 075108 (2024); and arXiv:2411.06167]