The question of non-normal returns in finance has a distinguished 90 year history. Mandelbrot (1963) brought it to prominance.
Mandelbrot himself advocated stable random variables with infinite variance. A decade later Peter Clark proposed time-change to handle non-normality based on volume clock. A very nice result following Clark was by Ane and Geman (2000) who showed that conditional on number of trades rather than Clark’s volume of trades one can obtain approximately normal distributions.
The state of the arts could be the CGMY model. My model is a time-fractional Heston model, which I expect to have advantages over the time-changed Levy models essentially because I handle the long memory aspect clearly. Long memory in volatility is a fundamental feature of all financial time series.