  Higher homotopies are nowadays playing a prominent role in mathematics as
well as in certain branches of theoretical physics. We recall some of the
connections between the past and the present developments. Higher homotopies
were isolated within algebraic topology at least as far back as the 1940's.
Prompted by the failure of the Alexander-Whitney multiplication of cocycles to
be commutative, Steenrod developed certain operations which measure this
failure in a coherent manner. Dold and Lashof extended Milnor's classifying
space construction to associative H-spaces, and a careful examination of this
extension led Stasheff to the discovery of An-spaces and Ainfty-spaces as
notions which control the failure of associativity in a coherent way so that
the classifying space construction can still be pushed through. Algebraic
versions of higher homotopies have, as we all know, led Kontsevich eventually
to the proof of the formality conjecture. Homological perturbation theory
(HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early
1950's, has nowadays become a standard tool to handle algebraic incarnations of
higher homotopies. A basic observation is that higher homotopy structures
behave much better relative to homotopy than strict structures, and HPT enables
one to exploit this observation in various concrete situations which, in
particular, leads to the effective calculation of various invariants which are
otherwise intractable. Higher homotopies abound but they are rarely recognized
explicitly and their significance is hardly understood; at times, their
appearance might at first glance even come as a surprise, for example in the
Kodaira-Spencer approach to deformations of complex manifolds or in the theory
of foliations.
