First year students learn that the Taylor expansion $Z_T$ carries
functions into power series, and that it has some nice algebraic
properties (e.g. multiplicativity, $Z_T(fg)=Z_T(f)Z_T(g)$). It is less
well known that the same game can be played within arbitrary groups:
there is a natural way to say "a Taylor expansion $Z$ for elements
of an arbitrary group $G$", and a natural way to carry the algebraic
properties of the Taylor expansion to this more general context. In the
case of a general $G$ "Taylor expansions" (expansions with the same
good properties as $Z_T$) may or may not exist, may or may not be unique,
may or may not separate group elements, and a further good property which
is hidden in the case of $Z_T$, "quadraticity", may or may not hold.
The purpose of this expository note is to properly define all the
notions in the above paragraph, to enumerate some classes of groups
whose theory of expansions we either understand or wish to understand,
to indicate the relationship between these notions and the
notions of "finite type invariants" and "unipotent" and "Mal'cev"
completions, and to point out (with references) that our generalization
of "expansions" to arbitrary groups is merely the tip of an iceberg,
for almost everything we say can be generalized further to "expansions
for arbitrary algebraic structures".