The validity of estimation and smoothing parameter selection for the wide class ofgeneralized additive models for location, scale and shape (GAMLSS) relies on the cor-rect specification of a likelihood function. Deviations from such assumption are knownto mislead any likelihood-based inference and can hinder penalization schemes meantto ensure some degree of smoothness for non-linear effects. We propose a generalapproach to achieve robustness in fitting GAMLSSs by limiting the contribution ofobservations with low log-likelihood values. Robust selection of the smoothing param-eters can be carried out either by minimizing information criteria that naturally arisefrom the robustified likelihood or via an extended Fellner-Schall method. The latterallows for automatic smoothing parameter selection and is particularly advantageousin applications with multiple smoothing parameters. We also address the challengeof tuning robust estimators for models with non-linear effects by proposing a novelmedian downweighting proportion criterion. This enables a fair comparison with ex-isting robust estimators for the special case of generalized additive models, whereour estimator competes favorably. The overall good performance of our proposal is illustrated by further simulations in the GAMLSS setting and by an application tofunctional magnetic resonance brain imaging using bivariate smoothing splines.