Functional regression is a dynamic research field within functional data analysis, with a wide range of applications across different areas. This paper aims to expand the existing framework of shape-constrained generalized additive models to functional generalized additive models with shape constraints. We introduce an extension of the shape-constrained P-spline (SCOP-spline) approach to a broad class of functional regression models with various shape constraints. Our framework includes parametric and a mixture of constrained and unconstrained smooth effects of functional and scalar covariates. Estimation and inference in this framework build upon the shape-constrained generalized additive models, enabling the use of wellestablished, robust, and flexible procedures. The described methods are implemented in the user-friendly R package scam. Simulation shows performance improvements when modelling with the proposed approach compared to the unconstrained method.