We introduce novel dynamical low-rank methods for solving large-scale matrix differential equa- tions, motivated by algorithms from randomized numerical linear algebra. In terms of perfor- mance (ratio accuracy/cost), our methods can overperform existing dynamical low-rank tech- niques. Several applications to stiff differential equations demonstrate the robustness, accuracy and low variance of the new methods, despite their inherent randomness. Allowing augmentation of the range and corange, the new methods have a good potential for preserving critical physical quantities such as the energy, mass and momentum. Numerical experiments on the Vlasov-Poisson equation are particularly encouraging. The new methods comprise two essential steps: a range estimation step followed by a post- processing step. The range estimation is achieved through a novel dynamical rangefinder method. Subsequently, we propose two methods for post-processing, leading to two time-stepping meth- ods: dynamical randomized singular value decomposition (DRSVD) and dynamical generalized Nyström (DGN). The new methods naturally extend to the rank-adaptive framework by estimat- ing the error via Gaussian sampling.