We study generalised additive models, with shape restrictions (e.g. monotonicity, convexity, concavity) imposed on each component of the additive prediction function. We show that this framework facilitates a nonparametric estimator of each additive component, obtained by maximising the likelihood. The procedure is free of tuning parameters and under mild conditions is proved to be uniformly consistent on compact intervals. More generally, our methodology can be applied to generalised additive index models. Here again, the procedure can be justified on theoretical grounds and, like the original algorithm, possesses highly competitive finite-sample performance. Practical utility is illustrated through the use of these methods in the analysis of two real datasets. Our algorithms are publicly available in the \texttt{R} package \textbf{scar}, short for \textbf{s}hape-\textbf{c}onstrained \textbf{a}dditive \textbf{r}egression.