Let 5 be a string of length N compressed into a context-free grammar S of size n. We present two representations of S achieving O(logN) random access time, and either O(n · αk(n)) construction time and space on the pointer machine model, or 0(n) construction time and space on the RAM. Here, αk(n) is the inverse of the kth row of Ackermann's function. Our representations also efficiently support decompression of any substring in S: we can decompress any substring of length m in the same complexity as a single random access query and additional O(m) time. Combining these results with fast algorithms for uncompressed approximate string matching leads to several efficient algorithms for approximate string matching on grammar-compressed strings without decompression. For instance, we can find all approximate occurrences of a pattern P with at most k errors in time O(n(min(|P|k,k4 + |P|) + logN) + occ), where occ is the number of occurrences of P in S. Finally, we are able to generalize our results to navigation and other operations on grammar-compressed trees. All of the above bounds significantly improve the currently best known results. To achieve these bounds, we introduce several new techniques and data structures of independent interest, including a predecessor data structure, two "biased" weighted ancestor data structures, and a compact representation of heavy-paths in grammars.