In the CLOSEST SUBSTRING problem k strings s1, . . . sk are given, and the task is to find a string s of length L such that each string si has a consecutive substring of length L whose distance is at most d from s. The problem is motivated by applications in computational biology. We present two algorithms that can be efficient for small fixed values of d and k: for some functions f and g, the algorithms have running time f(d)? n^o^{(\log d)} and g(d,k)?n^o ^{(\log log)},respectively. The second algorithm is based on connections with the extremal combinatorics of hypergraphs. The CLOSEST SUBSTRING problem is also investigated from the parameterized complexity point of view. Answering an open question from [6, 7, 11, 12], we show that the problem is W[1]- hard even if both d and k are parameters. It follows as a consequence of this hardness result that our algorithms are optimal in the sense that the exponent of n in the running time cannot be improved to o(logd) or to o(log log k) (modulo some complexity0-theoretic assumptions). Another consequence is that the running time n^o ^{(1/\varepsilon^4)} of the approximation scheme for CLOSEST SUBSTRING presented in [13] cannot be improved to f (\varepsilon) ? {n^c}, i.e., the \varepsilon has to appear in the exponent of n.