Adaptive spline models for geometric modeling and spline-based PDEs
solvers have recently attracted increasing attention both in the context
of computer aided geometric design and isogeometric analysis. In 
particular, approximation spaces defined over extensions of tensor-product 
meshes which allow axis aligned segments with T-junctions are currently 
receiving particular attention. Among others, the hierarchical approach 
is a relatively simple but powerful solution to address the problem of a 
local and adaptive mesh refinement in classical approximation algorithms.
Starting from the classical construction of hierarchical tensor-product 
B-splines, the talk will address recent results concerning the characterization
of multilevel spline spaces, the identification of suitable hierarchical 
bases and related topics.