What are the diseases that can be treated by gene and cell therapy currently?

In May 2019, the FDA approved the drug Zolgensma to treat spinal muscular atrophy in children under two years old. 

In 2017, the 1st gene therapy was approved in the EUA. The drug is Luxturna, used to treat genetic vision loss in patients with Leber congenital amaurosis or retinitis pigmentosa, both inherited retinal diseases.

Other drugs approved by the FDA are Yescarta, Kymriah, and Tecartus. These are CAR-T cell therapies used in the treatment of hematological cancers. 

References

https://www.asgct.org/education/more-resources/gene-and-cell-therapy-faqs