For any coprime integers gcd(a,b)=1 we consider the "rational Catalan number" Cat(a,b) = (a+b choose a) / (a+b). The classical Catalan numbers are Cat(n,n+1) and the Fuss-Catalan numbers are Cat(n,mn+1). Many common Catalan objects have "rational generalizations". The most basic are the "rational Dyck paths", which are lattice paths in an a by b rectangle staying above the diagonal. In this talk we will describe joint work with Brendon Rhoades and Nathan Williams on rational generalizations of noncrossing partitions and polygon triangulations (associahedra). We will also discuss joint work with Nick Loehr and Greg Warrington on a rational generalization of parking functions counted by b^{a-1}.