It is argued that the symmetry and anti-symmetry of the wave functions of systems consisting of ‘identical particles’ have nothing to do with the observational indistinguishability of these particles. Rather, a much stronger ‘conceptual indistinguishability’ is at the bottom of the symmetry requirements. This can be used to argue further, in analogy to old arguments of De Broglie and Schrödinger, that the reality described by quantum mechanics has a wave-like rather than particle-like structure. The question of whether quantum statistics alone can give rise to empirically observable correlations between results of distant measurements is also discussed.