Most causal inference methods consider counterfactuals under interventions that set the treatment deterministically. With continuous treatments or exposures, such counterfactuals may be of little practical interest because no feasible intervention would bring them about. Violations to the positivity assumption, necessary for identification, are exacerbated with continuous treatments and deterministic interventions. In this paper we propose longitudinal modified treatment policies (LMTPs) as a non-parametric alternative. LMTPs can always be guaranteed to satisfy positivity, and yield effects of immediate practical relevance with an interpretation that is familiar to regular users of linear regression adjustment. We study the identification of the LMTP parameter, study properties of the statistical estimand such as the efficient influence function, and propose four different estimators. Two of our estimators are efficient, and one is sequentially doubly robust in the sense that it is consistent if, for each time point, either an outcome regression or a treatment mechanism is consistently estimated. We perform a simulation study to illustrate the properties of the estimators.