Absolute neighborhood retracts (ANRs) are topological spaces $X$ which, whenever $i\colon X\to Y$ is an embedding into a normal topological space $Y$, there exists a neighborhood $U$ of $i(X)$ in $Y$ and a retraction of $U$ onto $i(X)$.
They were invented by Borsuk in 1932 (*Über eine Klasse von lokal zusammenhängenden Räumen*, Fundamenta Mathematicae **19** (1), p. 220-242, [EuDML](https://eudml.org/doc/212574)) and have been the object of a lot of developments from 1930 to the 60s (Hu's monograph on the subject dates from 1965),
being a central subject in combinatorial topology.


The discovery that these spaces had good topological (local connectedness),
homological (finiteness in the compact case) and even homotopical properties
must have been a strong impetus for the developement of the theory.
Also, they probably played some role in the discovery of the homotopy extension property
(it is easy to extend homotopies whose source is a normal space and
target an ANR) and of cofibrations.


I have the impression that this more or less gradually stopped being so in the 70s: a basic MathScinet search does not refer that many recent papers, although they seem to be used as an important tool in some recent works (a colleague pointed to me those of Steve Ferry).


My question (which does not want to be subjective nor argumentative) is the following:
what is the importance of this notion in modern developments of algebraic topology?



The role of ANR in modern topology
Another reason you might not see the word ANR these days is that compact finite-dimensional spaces are ANRs if and only if they are locally contractible. Thus, "finite-dimensional and local contractible" can replace ANR in the statement of a theorem (and might help the result appeal to a wider audience). 


In comparison geometry, for instance, the existence of a contractibility function takes the place of the ANR condition.


Borsuk conjectured that compact ANRs should have the homotopy types of finite simplicial complexes. Chapman and West proved that they even have preferred simple-homotopy types. This is part of the "topological invariance of torsion" package and is quite a striking result. Every compact, finite-dimensional, locally contractible space has a preferred finite combinatorial structure that is well-defined up to (even local!) simple-homotopy moves.


