  For decades, de Casteljau's algorithm has been used as a fundamental building
block in curve and surface design and has found a wide range of applications in
fields such as scientific computing, and discrete geometry to name but a few.
With increasing interest in nonlinear data science, its constructive approach
has been shown to provide a principled way to generalize parametric smooth
curves to manifolds. These curves have found remarkable new applications in the
analysis of parameter-dependent, geometric data. This article provides a survey
of the recent theoretical developments in this exciting area as well as its
applications in fields such as geometric morphometrics and longitudinal data
analysis in medicine, archaeology, and meteorology.
