We review the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, both on the classical and quantum level. Submitted to Reviews of Modern Physics.

We review recent work in which compactifications of string and M theory are constructed in which all scalar fields (moduli) are massive, and supersymmetry is broken with a small positive cosmological constant, features needed to reproduce real world physics. We explain how this work implies that there is a "landscape" of string/M theory vacua, perhaps containing many candidates for describing real world physics, and present the arguments for and against this idea. We discuss statistical surveys of the landscape, and the prospects for testable consequences of this picture, such as observable effects of moduli, constraints on early cosmology, and predictions for the scale of supersymmetry breaking.Let us briefly discuss the important physical scales in a compactification. Of course, one of the main goals is to explain the observed four dimensional Planck scale, which we denote M P,4 or simply M P . By elementary Kaluza-Klein reduction of D-dimensional supergravity,this will be related to the D-dimensional Planck scale M P,D , and the volume of the compactification manifold Vol(M ). Instead of the volume, let us define the Kaluza-Klein scale M KK = 1/V ol(M ) 1/(D−4) , at which we expect to see Kaluza-Klein excitations; the relation then becomes M 2 P,4 = M D−2 P,D M D−4 KK .(1)In the simplest (or "small extra dimension") picture, used in the original work on string compactification, all of these scales are assumed to be roughly equal. If the Yang-Mills sector is also D-dimensional, this is forced upon us, to obtain an order one four-dimensional gauge coupling; there are other possibilities as well.

We show that four-dimensional N = 2 supersymmetric SU (N ) gauge theory for N > 2 necessarily contains vacua with mutually non-local massless dyons, using only analyticity of the effective action and the weak coupling limit of the moduli space of vacua. A specific example is the Z 3 point in the exact solution for SU (3), and we study its effective Lagrangian. We propose that the low-energy theory at this point is an N = 2 superconformal U (1) gauge theory containing both electrically and magnetically charged massless hypermultiplets.

We show that configurations of multiple D-branes related by SU(N) rotations will preserve unbroken supersymmetry. This includes cases in which two D-branes are related by a rotation of arbitrarily small angle, and we discuss some of the physics of this. In particular, we discuss a way of obtaining 4D chiral fermions on the intersection of Dbranes.We also rephrase the condition for unbroken supersymmety as the condition that a 'generalized holonomy group' associated with the brane configuration and manifold is reduced, and relate this condition (in Type IIA string theory) to a condition in eleven dimensions.June 1996 (revised)

We give results for the distribution and number of flux vacua of various types, supersymmetric and nonsupersymmetric, in IIb string theory compactified on Calabi-Yau manifolds. We compare this with related problems such as counting attractor points.

Motivated by recent work of Dijkgraaf and Vafa, we study anomalies and the chiral ring structure in a supersymmetric U (N ) gauge theory with an adjoint chiral superfield and an arbitrary superpotential. A certain generalization of the Konishi anomaly leads to an equation which is identical to the loop equation of a bosonic matrix model. This allows us to solve for the expectation values of the chiral operators as functions of a finite number of "integration constants." From this, we can derive the Dijkgraaf-Vafa relation of the effective superpotential to a matrix model. Some of our results are applicable to more general theories. For example, we determine the classical relations and quantum deformations of the chiral ring of N = 1 super Yang-Mills theory with SU (N ) gauge group, showing, as one consequence, that all supersymmetric vacua of this theory have a nonzero chiral condensate.

We study the behavior of D-branes at distances far shorter than the string length scale l s .We argue that short-distance phenomena are described by the IR behavior of the Dbrane world-volume quantum theory. This description is valid until the brane motion becomes relativistic. At weak string coupling g s this corresponds to momenta and energies far above string scale. We use 0-brane quantum mechanics to study 0-brane collisions and find structure at length scales corresponding to the eleven-dimensional Planck lengths l s ) and to the radius of the eleventh dimension in M-theory (R 11 ∼ g s l s ). We use 0-branes to probe non-trivial geometries and topologies at sub-stringy scales. We study the 0-brane 4-brane system, calculating the 0-brane moduli space metric, and find the bound state at threshold, which has characteristic size l 11 P . We examine the blowup of an orbifold and are able to resolve the resulting S 2 down to size l 11 P . A 0-brane with momentum approaching 1/R 11 is able to explore a larger configuration space in which the blowup is embedded. Analogous phenomena occur for small instantons. We finally turn to 1-branes and calculate the size of a bound state to be ∼ g

We discuss a set of universal couplings between superstring Ramond-Ramond gauge fields and the gauge fields internal to D-branes, with emphasis on their topological consequences, and argue that instanton solutions in these internal theories are equivalent to D-branes. A particular example is the Dirichlet 5-brane in type I theory, which Witten recently showed is is the zero size limit of an SO(32) instanton. Its effective world-volume theory is an Sp(1) gauge theory, unbroken in the zero size limit. We show that the zero size limit of an instanton in this theory is a 1-brane, which can be described as a bound state of the Dirichlet 1-brane with the 5-brane. Considering several 1 and 5-branes provides a description of moduli spaces of Sp(N ) instantons, and a type II generalization is given which should describe U (N ) instantons.December 1995