This paper is a continuation of the first author's survey Local Euler Obstruction, Old and New (1998). It takes into account recent results obtained by various authors, in particular concerning extensions of the local Euler obstruction for frames, functions and maps and for differential forms and collections of them.
The local Euler obstruction was first introduced by R. MacPherson in  as a key ingredient for his construction of characteristic classes of singular complex algebraic varieties. Then, an equivalent definition was given by J.-P. Brasselet and M.-H. Schwartz in  using vector fields. This new viewpoint brought the local Euler obstruction into the framework of “indices of vector fields on singular varieties”, though the definition only considers radial vector fields. There are various other definitions and interpretations in particular due to Gonzalez-Sprinberg, Verdier, Lê-Teissier and others, and there is a very ample literature on this topic, see for instance  and also [1, 7, 9, 12, 13, 17, 23, 33, 34].
A survey was written by the first author . Then, the notion of local Euler obstruction developed mainly in two directions: the first one comes back to MacPherson's definition and concerns differential forms. That is developed by W. Ebeling and S. Gusein-Zade in a series of papers. The second one relates local Euler obstruction with functions defined on the variety [5, 6] and with maps . That approach is useful to relate local Euler obstruction with other indices.