This study focuses on the description of damage and failure behaviour of triaxial braided carbon/epoxy composites under tension. The tensile tests were instrumented with optical surface strain and acoustic emission measurements. Damage was observed using X-ray and microscopy. The damage develops in two stages: (1) intra-yarn cracking: increase of the crack density and crack length, (2) local inter-yarn delamination and conjunction of the intra-yarn cracks. Statistics of crack sizes at both stages were collected and the 3D geometry of cracks was reconstructed. A finite element model of the unit cell of the textile reinforcement is used to predict damage initiation and crack orientation using Puck’s criterion. Progressive damage and stiffness deterioration is modelled using the degradation scheme of Murakami–Ohno and the damage evolution law of Ladeveze, applied to the average stress state of the yarns. Good agreement with experimental damage initiation threshold and non-linear tensile diagrams is found both for loading in fibre and off-axis directions.