What is the best current summary of the limits of (tempered) distribution theory where it comes to multiplication of distributions? As I understand it, it has been shown that a consistent calculus of (tempered or otherwise) distributions involving multiplication cannot be constructed. Since the problem seems central to the theory of differential equations as a whole, quite a number of partial extensions of the theory exist, which sometimes permit multiplication of distributions as well. However, it seems those extensions take a number of different and mutually incomparable routes towards their aim. They do not seem to lead to a cohesive whole, at least quite at this moment.
So my question is, is there a generally thought to be comprehensive survey of the different approaches to the distributional multiplication problem, somewhere? Preferably one absent a paywall, for those of us such as I who linger in the periphery of the mathematical profession, and so do not have the means to consult serious literature, nor serious professionals in how to find the more spread out free stuff.
I would especially appreciate a treatment which intuitively motivates the hardness of the problem, and why so many different and disparate approaches have necessarily been tried towards even a partial solution. Any possible connections to formal mathematical logic would be highly useful to me as well.
 A: Great question!
I think the statement "However, it seems those extensions take a number of different and mutually incomparable routes towards their aim. They do not seem to lead to a cohesive whole, at least quite at this moment." is not totally accurate. There is in some sense a canonical procedure for multiplying distributions. The only thing is it does not always work. The various theorems in the literature give various sufficient conditions for this canonical procedure to work. Thus there is some unity between these results.
This procedure is as follows. Suppose we want to multiply two distributions $A$, $B$ in $\mathscr{D}'(\mathbb{R}^n)$. Let $\rho$ be a mollifier, i.e., a test function in $\mathscr{D}(\mathbb{R}^n)$ such that $\int \rho(x)\ d^nx=1$. For $\epsilon>0$
define the rescaled mollifier $\rho_{\epsilon}(x)=\epsilon^{-n}\rho(\frac{x}{\epsilon})$ and the functions $A_{\epsilon}=A\ast\rho_{\epsilon}$ and 
 $B_{\epsilon}=B\ast\rho_{\epsilon}$. Namely,
$$
A_{\epsilon}(x)=\langle A(y),\rho_{\epsilon}(x-y)\rangle_y
$$
which is just a notation for $A$ applied to the function $\rho_{\epsilon}(x-\bullet)$ or equivalently the function $y\mapsto \rho_{\epsilon}(x-y)$ with $x$ kept constant. $B_\epsilon$ is defined in the same way. It turns out the two functions $A_\epsilon$ and $B_\epsilon$ are $C^{\infty}$ so the product $A_{\epsilon}(x)B_{\epsilon}(x)$ is a well defined $C^{\infty}$ function and thus can be seen as an element $A_\epsilon B_\epsilon$ of $\mathscr{D}'(\mathbb{R}^n)$.
One can then ask, does the limit $\lim_{\epsilon\rightarrow 0}A_\epsilon B_{\epsilon}$ exist in $\mathscr{D}'(\mathbb{R}^n)$ equipped with the strong topology. If it does the procedure works. If not then the procedure fails.
A more interesting situation arises when trying to multiply random distributions. In that case, one typically considers $A_\epsilon B_{\epsilon}$ but subtracts some $\epsilon$ dependent constants (or other previously constructed products) before taking the limit $\epsilon\rightarrow 0$. 
Below are some references to learn more.
The best place to start, would be the introduction of reference 1 in order to get an idea of the literature. Then to get into the subject itself,
reference 2 is a pedagogical introduction with several explicit examples, about Hörmander's Theorem for the multiplication of distributions.
In the bibliography of reference 1 you will find the original article by Schwartz on the multiplication of distributions as well as to the book by Bahouri, Chemin and Danchin which contains an approach via paraproducts. For the random case you can see references 1 and 3.
References:


*

*My article "A Second-Quantized Kolmogorov–Chentsov Theorem via the Operator Product Expansion", Comm. Math. Phys. 376 (2020), 555–608. See here for the free preprint version.

*C. Brouder, N. V. Dang, and F. Hélein, "A smooth introduction to the wavefront set", J. Phys. A 47 (2014), 44.
See here for the free preprint version.

*A. Chandra, and H. Weber, "Stochastic PDEs, Regularity structures, and interacting particle systems", Ann.  Fac. des sciences de Toulouse : Math. (ser. 6), 26 (2017), 847-909. See  here for the free preprint version.

