We prove that sectional-hyperbolic attracting sets for C 1 vector fields are robustly expansive (under an open technical condition of strong dissipativeness for higher codimensional cases). This extends known results of expansiveness for singular-hyperbolic attractors in 3-flows even in this low dimensional setting. We deduce some converse results taking advantage of recent progress in the study of star vector fields: a robustly expansive non-singular vector field is uniformly hyperbolic; and a robustly transitive attractor is sectional-hyperbolic if, and only if, it is robustly expansive. In a low dimensional setting, we show that an attracting set of a 3-flow is singular-hyperbolic if, and only if, it is robustly chaotic (robustly sensitive to initial conditions). CONTENTS 1. Introduction 2. Statement of the results 2.1. Sectional-hyperbolic attracting sets 2.2. Robust expansiveness for codimension-two sectional-hyperbolic attracting sets 2.3. Robust expansiveness for higher codimension 2.4. Some consequences of robust expansiveness 2.5. Organization of the text Acknowledgements 3. Preliminary results on sectional-hyperbolic attracting sets 3.1. Generalized Lorenz-like singularities 3.2. Invariant extension of the stable bundle 3.3. Extension of the center-unstable cone field 3.4. Global Poincaré map on adapted cross-sections 3.5. Hyperbolicity of the global Poincaré map 3.6. Distance between points on distinct stable leaves in cross-sections 3.7. The Poincaré quotient map on a cross-section 4. Proof of robust expansiveness 4.1. Proof of the main Theorems A and B 4.2. Proof of positive expansiveness 4.3. Proof of the claim

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