We introduce $H$-join decompositions of graphs, indexed by a fixed bipartite graph $H$. These decompositions are based on a graph operation that we call $H$-join, which adds edges between two given graphs by taking partitions of their two vertex sets, identifying the classes of the partitions with vertices of $H$, and connecting classes by the pattern $H$. $H$-join decompositions are related to modular, split and rank decompositions. Given an $H$-join decomposition of an $n$-vertex $m$-edge graph $G$ we solve the Maximum Independent Set and Minimum Dominating Set problems on $G$ in time $O(n(m+2^{O(\rho(H)^2)}))$, and the $q$-Coloring problem in time $O(n(m+2^{O(q\rho(H)^2)}))$, where $\rho(H)$ is the rank of the adjacency matrix of $H$ over GF(2). Rankwidth is a graph parameter introduced by Oum and Seymour, based on ranks of adjacency matrices over GF(2). For any positive integer $k$ we define a bipartite graph $R_k$ and show that the graphs of rankwidth at most $k$ are exactly the graphs having an $R_k$-join decomposition, thereby giving an alternative graph-theoretic definition of rankwidth that does not use linear algebra. Combining our results we get algorithms that, for a graph $G$ of rankwidth $k$ given with its width $k$ rank-decomposition, solves the Maximum Independent Set problem in time $O(n(m+2^{\frac{1}{2} k^2+\frac{9}{2} k} \times k^2 ))$, the Minimum Dominating Set problem in time $O(n(m+2^{\frac{3}{4} k^2+\frac{23}{4} k} \times k^3))$ and the $q$-Coloring problem in time $O(n(m+2^{\frac{q}{2} k^2+\frac{5q+4}{2} k} \times k^{2q} \times q ))$. These are the first algorithms for NP-hard problems whose runtimes are single exponential in the rankwidth.