Description:    It has been conjectured since the work of Lalley and Sellke (Ann. Probab., 15, 1052–1061, 1987 ) that branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be proved in several different ways (see e.g. Brunet and Derrida, A branching random walk seen from the tip, 2010 , Poissonian statistics in the extremal process of branching Brownian motion, 2010 ; Arguin et al., The extremal process of branching Brownian motion, 2011 ). The structure of this extremal point process turns out to be a Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process. The main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process. Another proof and description has been obtained independently by Arguin et al. (The extremal process of branching Brownian motion, 2011 ). Content Type Journal Article Pages 1-47 DOI 10.1007/s00440-012-0461-0 Authors E. Aïdékon, Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, Eindhoven, The Netherlands J. Berestycki, Laboratoire de Probabilités et Modèles Aléatoires, CNRS UMR 7599, UPMC Université Paris 6, Case courrier 188, 4, Place Jussieu, 75252  Paris Cedex 05, France É. Brunet, Laboratoire de Physique Statistique, École Normale Supérieure, UPMC Université Paris 6, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005  Paris, France Z. Shi, Laboratoire de Probabilités et Modèles Aléatoires, CNRS UMR 7599, UPMC Université Paris 6, Case courrier 188, 4, Place Jussieu, 75252  Paris Cedex 05, France Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051