Li-Chung Chen and Mark Haiman studied a family of symmetric functions called Catalan (symmetric) functions which are indexed by pairs consisting of a partition contained in the staircase $(n-1, \dots, 1, 0)$ (of which there are Catalan many) and a composition weight of length $n$. They include the Schur functions, the Hall-Littlewood polynomials and their parabolic generalizations. They can be defined by a Demazure-operator formula, and are equal to the $GL$-equivariant Euler characteristics of vector bundles on the flag variety by the Borel-Weil-Bott Theorem. We have discovered various properties of Catalan functions, providing new insight on the existing theorems and conjectures inspired by the Macdonald Positivity Conjecture. A key discovery in our work is an elegant set of ideals of roots whose associate Catalan functions are $k$-Schur functions, proving that graded $k$-Schur functions are $GL$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We exposed a new shift invariance property of the graded $k$-Schur functions and resolved the Schur positivity and $k$-branching conjectures by providing direct combinatorial formulas using strong marked tableaux. We conjectured that Catalan functions with a partition weight are $k$-Schur positive which strengthens the Schur positivity of the Catalan function conjecture by Chen-Haiman and resolved the conjecture with positive combinatorial formulas in cases which capture and refine a variety of problems. This is joint work with Jonah Blasiak, Jennifer Morse, and Daniel Summers.