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String duality suggests a fascinating juxtoposition of world-volume and target-space dynamics. This is particularly apparent in the D-brane description of stringy solitons that forms a major focus of this article (which is not intended to be a comprehensive review of this extensive subject). The article is divided into four sections: • The oligarchy of string world-sheets • p-branes and world-volumes • World-sheets for world-volumes • Boundary states, D-branes and space-time supersymmetry [email protected] This article is based on a talk presented at the CERN Workshop on String Duality (December, 1995) and the published version of a talk at the Buckow Symposium (September, 1995). 1 The oligarchy of string world-sheets According to current lore all known superstring perturbation expansions may be viewed as different approximations to a single underlying theory. Viewed from the standpoint of any given string theory the fundamental strings of other theories are BPS solitons that are not apparent as particle states in string perturbation theory. In addition there are other solitonic solutions with p-dimensional spatial extension which play a vital rôle in the duality symmetries of the theory. All of these solitons have the property that they preserve some fraction of space-time supersymmetry so that they are analogous to the BPS monopoles in the Higgs–Yang–Mills system. The interrelations between different perturbation expansions [1, 2] (and references therein) appear to be profound and should indicate the path towards a more fundamental way of expressing the theory in which no particular perturbative approximation has preferred status. However, strings (p = 1) do have an exalted position among this panoply of solitons. Only those theories in which the fundamental states are strings possess well-defined (albeit non-convergent) world-volume and target-space perturbation expansions. Thus, although first-quantized point-particle (p = 0) quantum mechanics is well-defined, a theory including gravity cannot be second-quantized in perturbation approximation. Moreover, p-brane quantum mechanics with p > 1 is ill-defined – the world-volume theories for such p-branes are non-renormalizable quantum theories and can only be interpreted as effective theories, suitable for describing long wavelength behaviour. But, following recent developments it has become plausible that the (p + 1)dimensional world-volume field theories of p-branes should be interpreted as effective low-energy approximations to an underlying string theory. This is implicit in the D-brane description of p-branes carrying the charges of the Ramond–Ramond (R⊗R ) sector [3, 4] (and a multitude of other papers) and plausibly (but more speculatively) of the solitonic heterotic string and fivebrane [5] which carry Neveu–Schwarz Neveu–Schwarz (NS⊗NS ) charges. This represents the implementation of the idea that the string world-sheet of one string theory (the ‘effective’ theory) might be interpreted as an effective two-dimensional target-space of another string theory (the ‘underlying’ theory) [6]. The world-sheet coordinates of the effective theory are world-sheet fields of the underlying theory while the world-sheet fields of the effective theory are target-space fields of the underlying theory. The string tension of the effective theory (which is the effective world-sheet loop counting parameter) is then determined in terms of the target-space closed-string coupling constant, g, of the underlying theory. If the underlying string theory is a closed-string theory the effective string tension behaves as 1/g while if it is an open-string theory the effective string tension behaves as 1/g. In this way there is a kind of ‘world-sheet democracy’ among string theories. It is natural to extend this idea to the description of p-branes with the (p + 1)-