Summary: Each positive rational number \(x>0\) can be written uniquely as \(x=a/(b-a)\) for coprime positive integers \(0<a<b\). We will identify \(x\) with the pair \((a,b)\). In this paper we define for each positive rational \(x>0\) a simplicial complex \(\mathsf{Ass}(x)=\mathsf{Ass}(a,b)\) called the rational associahedron. It is a pure simplicial complex of dimension \(a-2\), and its maximal faces are counted by the rational Catalan number \[ \mathsf{Cat}(x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)!}{a!\,b!}. \] The cases \((a,b)=(n,n+1)\) and \((a,b)=(n,kn+1)\) recover the classical associahedron and its “Fuss-Catalan” generalization studied by C. A. Athanasiadis and E. Tzanaki [Isr. J. Math. 167, 177–191 (2008; Zbl 1200.05252)] and S. Fomin and N. Reading [Int. Math. Res. Not. 2005, No. 44, 2709–2757 (2005; Zbl 1117.52017)]. We prove that \(\mathsf{Ass}(a,b)\) is shellable and give nice product formulas for its \(h\)-vector (the rational Narayana numbers) and \(f\)-vector (the rational Kirkman numbers). We define \(\mathsf{Ass}(a,b)\) via rational Dyck paths: lattice paths from \((0,0)\) to \((b,a)\) staying above the line \(y = \frac{a}{b}x\). We also use rational Dyck paths to define a rational generalization of noncrossing perfect matchings of \([2n]\). In the case \((a,b) = (n, mn+1)\), our construction produces the noncrossing partitions of \([(m+1)n]\) in which each block has size \(m+1\).