Known by many names, sketching techniques allow random projections of data from high to low dimensions while preserving pairwise distances. This talk explores ways to use sketching so as to improve the scalability of algorithms for diverse classes of optimization problems and applications, from linear to nonlinear, local to global, derivative-based to derivative-free. Regression problems and Gauss-Newton techniques will receive particular attention. Numerical illustrations on standard optimization test problems as well as on some machine learning set-ups will be presented. This work is joint with Jan Fiala (NAG Ltd), Jaroslav Fowkes (Oxford), Estelle Massart (Oxford and NPL), Adilet Otemissov (Oxford and Turing), Alex Puiu (Oxford), Lindon Roberts (Australian National University, Canberra), Zhen Shao (Oxford).