Polynomial splines on unstructured triangulations in finite element theory have been traditionally characterized by interpolation problems such that each polynomial piece is determined by local interpolation data. To align these techniques with the concepts of isogeometric analysis, one approach is to express finite elements in the Bernstein-Bezier representation. A further advance in this direction is the development of globally defined locally supported basis functions on triangulations that form a convex partition of unity, thereby mimicking the properties of tensor product B-splines. One obvious benefit of this framework is the possibility to define well-behaved rational basis functions on unstructured partitions. In this contribution we review the techniques for constructing B-spline-like functions on triangulations and consider their natural extension to rational forms by assigning a positive weight to each basis function. We discuss domain parametrization methods and conversion from NURBS. In addition, we present numerical examples of solving boundary value problems on planar and surface domains and demonstrate options for local refinement.