The algebra of multivariate symmetric polynomials has been used by mathematicians for hundreds of years and contains rich combinatorial structures. One particularly rich and classical interaction occurs between symmetric polynomials and representation theory. More recently, an even deeper program has developed for understanding graded symmetric group representations by enriching the coefficients of the symmetric polynomials to include formal parameters. A foundational example of such a result is the "Shuffle theorem", which gives a combinatorial formula for a symmetric polynomial associated to the ring of diagonal coinvariants of the symmetric group via the combinatorics of Dyck paths. This result was originally conjectured in 2005 and was not resolved until 2018. In the intervening years, a rich theory of graded combinatorics was developed with many algebraic connections and conjectures. In recent work, my collaborators and I have proved various generalizations of these shuffle theorems by developing a toolkit to bridge between these kinds of algebraic and combinatorial expressions, using a tool we call "Catalanimals." In this talk, I will give a flavor of these combinatorial results and some motivations for studying them.