© 2020 Royal Statistical Society We introduce a general, flexible, parametric survival modelling framework which encompasses key shapes of hazard functions (constant; increasing; decreasing; up then down; down then up) and various common survival distributions (log-logistic; Burr type XII; Weibull; Gompertz) and includes defective distributions (cure models). This generality is achieved by using four distributional parameters: two scale-type parameters¿one of which relates to accelerated failure time (AFT) modelling; the other to proportional hazards (PH) modelling¿and two shape parameters. Furthermore, we advocate `multiparameter regression¿ whereby more than one distributional parameter depends on covariates¿rather than the usual convention of having a single covariate-dependent (scale) parameter. This general formulation unifies the most popular survival models, enabling us to consider the practical value of possible modelling choices. In particular, we suggest introducing covariates through just one or other of the two scale parameters (covering AFT and PH models), and through a `power¿ shape parameter (covering more complex non-AFT or non-PH effects); the other shape parameter remains covariate independent and handles automatic selection of the baseline distribution. We explore inferential issues and compare with alternative models through various simulation studies, with particular focus on evidence concerning the need, or otherwise, to include both AFT and PH parameters. We illustrate the efficacy of our modelling framework by using data from lung cancer, melanoma and kidney function studies. Censoring is accommodated throughout.