Abstract: Over the last decades, the well-posedness issue of stochastic dispersive PDEs with multiplicative noises has been extensively studied. However, this study was done primarily from the viewpoint of Ito solution theory, and pathwise well-posedness remained completely open. In this talk, I will present the first pathwise well-posedness results for stochastic nonlinear wave equations (SNLW) and stochastic nonlinear Schrödinger equations (SNLS) with multiplicative white-in-time/coloured-in-space noise. Here, we combine the operator-value controlled rough paths adapted to dispersive flows, together with random tensor estimates, and the Fourier restriction norm method adapted to controlled rough paths.