In 1965 Penrose introduced the fundamental concept of a trapped surface, on the basis of which he proved a theorem which asserts that a spacetime containing such a surface must come to an end. The presence of a trapped surface implies, moreover, that there is a region of spacetime, the black hole, which is inaccessible to observation from infinity. Since that time a major challenge has been to find out how trapped surfaces actually form, by analyzing the dynamics of gravitational collapse. The present monograph achieves this aim by establishing the formation of trapped surfaces in pure general relativity through the focusing of gravitational waves. The theorems proved in this monograph constitute the first foray into the longtime dynamics of general relativity in the large, that is, when the initial data are no longer confined to a suitable neighborhood of trivial data. The main new method, the short pulse method, applies to general systems of EulerLagrange equations of hyperbolic type and provides the means to tackle problems which have hitherto seemed unapproachable. This monograph will be of interest to people working in general relativity, geometric analysis, and partial differential equations. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in mathematical physics. Table of Contents  The optical structure equations
 The characteristic initial data
 \(L^\infty\) estimates for the connection coefficients
 \(L^4(S)\)estimates for the 1st derivatives of the connection coefficients
 The uniformization theorem
 \(L^4(S)\) estimates for the 2nd derivatives of the connection coefficients
 \(L^2\) estimates for the 3rd derivatives of the connection coefficients
 The multiplier fields and the commutation fields
 Estimates for the derivatives of the deformation tensors of the commutation fields
 The Sobolev inequalities on the \(C_u\) and the \({\underline C}_{\underline u}\)
 The \(S\)tangential derivatives and the rotational Lie derivatives
 Weyl fields and currents. The existence theorem
 The multiplier error estimates
 The 1storder Weyl current error estimates
 The 2ndorder Weyl current error estimates
 The energyflux estimates. Completion of the continuity argument
 Trapped surface formation
 Bibliography
 Index
