Description:    We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the clusters is a Poisson process with intensity measure with exponential density. The law of the individual clusters is characterized as branching Brownian motions conditioned to perform “unusually large displacements”, and its existence is proved. The proof combines three main ingredients. First, the results of Bramson on the convergence of solutions of the Kolmogorov–Petrovsky–Piscounov equation with general initial conditions to standing waves. Second, the integral representations of such waves as first obtained by Lalley and Sellke in the case of Heaviside initial conditions. Third, a proper identification of the tail of the extremal process with an auxiliary process (based on the work of Chauvin and Rouault), which fully captures the large time asymptotics of the extremal process. The analysis through the auxiliary process is a rigorous formulation of the cavity method developed in the study of mean field spin glasses. Content Type Journal Article Pages 1-40 DOI 10.1007/s00440-012-0464-x Authors Louis-Pierre Arguin, Département de Mathématiques et de Statistique, Université de Montréal, 2920 Chemin de la Tour,  Montréal,  H3T 1J4 Canada Anton Bovier, Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany Nicola Kistler, Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051