This talk is supposed to provide a panoramic view on different ways of defining the convolution of function, pseudo-measures or distributions, starting from the very classical setting of the Lebesgue space $l^1(R^d)$ where it (still) can be defined in the pointwise sense (almost everywhere), thus turning this Banach space into a commutative Banach algebra with bounded approximate units. This is often taken as a starting point for Fourier Analysis (in particular for the study of the question of spectral analysis via closed ideals of $L^1$,
as outlined in the book of Hans Reiter from 1968 end elsewhere).

There have been many attempts to extend this notion beyond the Lebesgue setting, e.g. for bounded measures, or for pseudo-measures (the elements of the
space $FL^\infty$ in a distributional setting), where one can resort to the pointwise multiplication on the Fourier transform side. On the other hand there have been long-standing attempts to define (at an individual level) the convolution of distributions, with the serious drawback that one may loose the expected rules of associativity.

As a short summary one can say that it is better to avoid pointwise considerations for the definition of convolution, and better connect the possible definitions of convolution with a distributional setting, e.g. in the context of mild distributions. Moreover, most if the time good definitions depend on (or can at least be related to) the identification of the distributions which ``can be convolved with each other'' with corresponding convolution operators between well defined (Banach) spaces of operators, where composition of operators makes sense. Commutativity of convolution can then often be derived via the strong operator topology for such convolution operators by more conventional convolution kernels, e.g. by test functions.

If time permits we will also shortly discuss the new approach to integrated group actions promoted by the author, which allows to introduce the definition of convolution of bounded measures over LCA groups plus the derivation of the convolution theorem (the Fourier-Stieltjes transform converts convolution into pointwise multiplication of bounded and continuous functions on the frequency domain) without the use of classical measure theory (rather by introducing bounded measures as linear functionals on $C_0(G)$), by identifying this dual space with the space of ``multipliers'', i.e. bounded linear operators commuting with translations (so-called BIBOS in the engineering literature).