In commonly used functional regression models, the regression of a scalar or functional response on the functional predictor is assumed to be linear. We relax the linearity assumption and propose to replace it by an additive structure. This leads to a more widely applicable and flexible framework for functional regression models. The regularization needed for effective estimation of the regression parameter function is implemented through a projection on the eigenbasis of the covariance operator of the functional components in the models.  The utilization of functional principal components in an additive rather than linear way leads to substantial broadening of the scope of functional regression models and emerges as a natural approach, as the uncorrelatedness of the functional principal components is shown to lead to a straightforward implementation of the functional additive model, just based on a sequence of one-dimensional smoothing steps and without need for backfitting. We further extend this framework to the estimation of  functional gradients which are then defined as functional directional derivatives. Aiming at a model-free approach, we are enabled to study how changes in the predictor function in a specified functional direction are associated with corresponding changes in the response. The proposed modeling framework and estimation methods are illustrated in applications to gene expression time course data and meflies data, respectively.