Starting from some examples, I will review various implications of classical consistency that have played a key role in the context of higher-spin theories (Vasiliev's equations) and holography. I will then review the current understanding higher-spin interactions involving totally-symmetric higher-spin fields and the associated classification of higher-spin algebras generalizing Coleman-Mandula theorem in the context of AdS/CFT. Finally, I will discuss the extension of this analysis to massive higher spins, towards a new understanding of string theory.