“There is sufficient evidence for the preclinical effectiveness and the clinical efficacy of homeopathy and for its safety and economy compared with conventional treatment and medicine” Swiss Government’s report

The main conditions with the most scientific research done until today are allergies, arthritis, respiratory conditions, ADHD among others. There are in vitro and in vivo studies that are published in renowned scientific journals like the Lancet, Scientific American etc. You can find the complete list here.