Twenty years of P-splines

P-splines first appeared in the limelight twenty years ago. Since then they have become popular applications and in theoretical work. The combination of a rich B-spline basis and a simple dif- penalty lends itself well to a variety of generalizations, because it is based on regression. effect, P-splines allow the building of a “backbone” for the “mixing and matching” of a variety additive smooth structure components, while inviting all sorts of extensions: varying-coefficient , signal (functional) regressors, two-dimensional surfaces, non-normal responses, quantile (expectile) modelling, among others. Strong connections with mixed models and Bayesian analy- have been established. We give an overview of many of the central developments during the two decades of P-splines.