What are the current theoretical solutions to the Hubble Tension? As it has been known in the last few years, there is a discrepancy in the measured values of the Hubble constant, depending on whether the method addresses early or late times of the cosmological history. Given the growing significance, (reviews here and here ) I believe theoreticians have started to take the tension more and more seriously. I am interested in getting to know what models are out there addressing the Hubble tension, a short list with a very rough explanation would be highly appreciated.
 A: There are several ways to solve or relax the Hubble tension. I'll try to provide the key idea and why it may work, then I'll provide a list of references that can help you go through the literature. The list won't be complete but from there you can easily find more approaches.
So in general, to relax the Hubble tension you can modify early time physics, late time physics or both. Late time physics is strongly constrained by BAO (Baryon Acustic Oscillations) so it is way more complicated to do that while staying consistent with observations. For this reason most of the models and this answer focus on early time solutions to the Hubble tension.
The majority of these models try to solve the tension in the same way, i.e. by reducing the sound horizon at the time of radiation drag while keeping fixed its angular size. The angular size has to be unchanged because it is observed with great precision by CMB experiments. (If you don't know what is baryon drag don't worry, you can think of it as the time of last scattering for our purposes).
So, how is it possible to decrease the sound horizon without modifying its angular size? Easy, by increasing the Hubble constant today! So, the sound horizon at baryon drag is given by
$$
r_s (z_d) = \int_{z_d}^{\infty} \frac{dz}{H(z)} c(z), 
$$
where, $z_d$ is the redshift of baryon drag while c(z) is the speed of sound in the photon-baryon fluid. So if you find a way to reduce $r_s (z_d)$ you have, at the same time to be able to keep fixed its angular size, which is observed, and it's given by
$$
\theta_s = \frac{r_s(z_d)}{D_A(z_d)}, 
$$
in which, the angular diameter distance, $D_A(z_d)$ is
$$
D_A(z_d) = \int_0^{z_d} \frac{dz}{H(z)}.
$$
So, from these formulas you can see that, if we are able to reduce $r_s(z_d)$, in order to keep $\theta_s$ fixed, we have to reduce $D_A(z_d)$ by the same amount. This is done by an higher Hubble parameter $H$ at late times, which means higher $H_0$
Some literature you might want to read
First of all, the first paper I think you should read to get a feel of the Hubble tension is, in my opinion, The trouble with H0. It is from 2016 so it's not up to date with the latest measurements but all the key aspects are explained very clearly.
Now, some actual approaches that try to solve the tension:

*

*Early dark energy (EDE): EDE1 EDE2, EDE3, EDE4, EDE5, EDE6, see also this seminar by Kamionkowski on the topic, it covers both an introduction to the Hubble tension and a possible solution using EDE.

*Extra radiation in either neutrinos Neff1, Neff2, Neff3 or some other dark sector DS1, DS2, and dark energy-dark matter interactions DMDE.

*Modifications to General Relativity. This is too vast to provide enough references here, not all modifications of gravity aims to solve this problem but in some models this happens naturally. In the context of scalar tensor theories I'm citing just two papers: MG1 and MG2 but there are many more and if you're interested you should look at references in these papers. You might also want to look at some reviews on scalar-tensor theories (google Horndeski theories review) and other modified gravity theories (there are many) that might help solve the tension.

*Primordial magnetic fields  PMF1.

*Non standard recombination NSR1.

*Varying fundamental constants FundConst1, FundConst2.

A: @RenatoRenatoRenato's answer is correct, however, if you also want a summary:

While no specific proposal makes a strong case for being highly likely
or far better than all others, solutions involving early or dynamical
dark energy, neutrino interactions, interacting cosmologies,
primordial magnetic fields, and modified gravity provide the best
options until a better alternative comes along

This is from a huge but quite readable paper (5 June 2021 - In the Realm of the Hubble Tension - a review of Solutions) that has reviewed and analysed a wide swath of the latest solution attempts.
