Tchebycheffian splines are smooth piecewise functions whose pieces are drawn from (possibly different) Tchebycheff spaces, a natural generalization of algebraic polynomial spaces. They enjoy most of the properties known in the polynomial spline case. In particular, under suitable assumptions, Tchebycheffian splines admit a representation in terms of basis functions, called Tchebycheffian B-splines (TB-splines), completely analogous to polynomial B-splines. A particularly interesting subclass consists of Tchebycheffian splines with pieces belonging to null-spaces of constant-coefficient linear differential operators. They grant the freedom of combining polynomials with exponential and trigonometric functions with any number of individual shape parameters. Moreover, they have been recently equipped with efficient evaluation and manipulation procedures. TB-splines offer an alternative to standard polynomial B-splines and NURBS in isogeometric methods. Local refinement strategies proposed for the polynomial case can be extended to the Tchebycheff setting. Structure preserving isogeometric discretizations based on TB-splines can be efficiently designed as well. It turns out that TB-splines can outperform polynomial B-splines whenever appropriate problem-driven selection strategies for the underlying Tchebycheff spaces are applied . In this talk we discuss some recent results about the use of TB-splines in the isogeometric paradigm. The talk is based on joint works with Krunal Raval, Hendrik Speleers, and Deepesh Toshniwal.