The aim of this talk is to extract the quadratic corrections to Fronsdal equations from Vasiliev's equations, and discuss their structure. The issue of locality in higher-spin theories will be addressed. Implications of this analysis in relation to Giombi-Yin checks of higher-spin/vector model duality will be discussed. Some of the previous puzzles will be resolved, while others will be elucidated.

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.