We introduce continuously additive models, which can be viewed as extensions of additive regression models with vector predictors to the case of infinite-dimensional predictors. This approach produces a class of flexible functional nonlinear regression models, where random predictor curves are coupled with scalar responses. In continuously additive modelling, integrals taken over a smooth surface along graphs of predictor functions relate the predictors to the responses in a nonlinear fashion. We use tensor product basis expansions to fit the smooth regression surface that characterizes the model. In a theoretical investigation, we show that the predictions obtained from fitting continuously additive models are consistent and asymptotically normal. We also consider extensions to generalized responses. The proposed class of models outperforms existing functional regression models in simulations and real-data examples.