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) 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?