Exact solutions of closed string theory

We review explicitly known exact $D=4$ solutions with Minkowski signature in closed bosonic string theory. Classical string solutions with space-time interpretation are represented by conformal sigma models. Two large (intersecting) classes of solutions are described by gauged WZW models and `chiral null models' (models with conserved chiral null current). The latter class includes plane-wave type backgrounds (admitting a covariantly constant null Killing vector) and backgrounds with two null Killing vectors (e.g., fundamental string solution). $D>4$ chiral null models describe some exact $D=4$ solutions with electromagnetic fields, for example, extreme electric black holes, charged fundamental strings and their generalisations. In addition, there exists a class of conformal models representing axially symmetric stationary magnetic flux tube backgrounds (including, in particular, the dilatonic Melvin solution). In contrast to spherically symmetric chiral null models for which the corresponding conformal field theory is not known explicitly, the magnetic flux tube models (together with some non-semisimple WZW models) are among the first examples of solvable unitary conformal string models with non-trivial $D=4$ curved space-time interpretation. For these models one is able to express the quantum hamiltonian in terms of free fields and to find explicitly the physical spectrum and string partition function.