A quick Google search on "psychedelic therapy" brings up a ton of results. It was originally legal and LSD was a major player and seen as a wonder drug in therapy before it was hastily banned after it exploded in popularity outside of the therapy scene.
MDMA and Psilocybin (one of the two major active components of Psilocybin mushrooms, the other being Psilocin, for which Psilocybin is a pro-drug) are both being studied currently for their use in therapy, with extremely promising results. MDMA in particular has been used in PTSD studies, and seems to be allowing people who previously couldn't talk about the major traumas they suffered to open up and relate to their therapists, getting things off their chest they had buried deep for many years, finally allowing them to heal. Psilocybin has shown promise in treating depression and anxiety too.
The results are quite astounding honestly even to someone like me who first-hand experienced what psychedelic therapy can do when I was able to get past my fear of people and socialising through a single alpha-Methyltryptamine therapy session (even with someone who'd never previously ingested a single psychedelic drug being my guide!). They're being picked up by the media and psychedelics and empathogens are finally being given some of the credit they deserve.
To me a cure is not a substance that you must take day by day for weeks, months, or years to become "stabilised" - a cure is something one takes once, that rids them of their ailment, and this is something that in many cases psychedelics have the potential of doing - with psychological disorders previously treated only with the likes of things like SSRIs, drugs that are addictive, have many side effects, and are prescribed for daily use.
While I think there has been some untoward negative light shone on psychedelics over the years since the times the first of them were banned, I think that people can see the clear evidence in their past history in psychotherapy, and that this isn't even a case like Cannabis where even many of the medical advocates were just looking for a legal loophole for legal personal use - it's a case where genuine powerful medicines have been put under lock and key for a long time and scientists have realised that may have been a tad rash, and so they're moving forward and taking a fresh look at them, and re-affirming they have their use.
So yes, I see it happening over the next few years.
While it's a dissociative rather than a typical psychedelic, Ketamine is already used legally to treat depression on a small scale, and MDMA and Psilocybin are being used in therapy as we speak in pre-emptive studies. Things will progress from there.
MDMA and Psilocybin are more than enough to cover most anything any other psychedelic or empathogen could treat, so I'll be happy just to see those two hit therapy by storm and remove some of the stigma surrounding them, and maybe help even some of those people that once thought they were right to be placed in the same drug legal schedules as the likes of Heroin. Be patient and science will do what it does and get things going.
I hope AMT is looked at too, though I feel its soviet origin and being less commonly used may make it a less likely choice, its pharmacology combining the effects and treatment potential of both MDMA and psychedelics (...with a little personal bias from my own experience thrown in
) makes it a candidate that could fit both shoes.