I think one has to distinguish between pricing and fitting/reproducing empirical stock returns. A model might fit the empirical stock returns extremly well but fail to reproduce derivative prices. In my answer I will assume that you are interested in reproducing the empirical stock returns.
Mandelbrot and the Stable Paretian Hypothesis
The most salient difference between the Normal distribution and the real world data are the heavy tails.
This has been first noted (or at least publicly asserted) by Bernoit Mandelbrot and resulted in his Stable Paretian Hypothesis. Mandelbrot argued that contrary to the Gaussian-Haypothesis the variance of stock returns can be infinite resulting in heavy tails. He proposes the use of stable distributions with infinite variance - that also known as stable paretian distributions. (for more on the stable paretian distributions see once again the paper on elaborating on the Stable Paretian Hypothesis)
Thus most contemporary approaches focuse on designing distributions (or processes with transition densities) that feature heavy tails.
First of all I don't think "the one" distribution for stock returns actually exists. Heavy-Tail ones will in most cases outperform the Normal distribution. The concrete choice may depend on the data one is working with. Still, there is a couple of examples worth mentioning.
The Heston-Model does a decent job in capturing the high kurtosis and the fat tails of stock returns. The paper Goodness-of-fit of the Heston model provides a detailes analysis. As it turns out the Heston-Model is well suited for reproducing daily and up to monthly returns but doesn't perform as well with annual stock returns. The original paper on the transition density of the HM can be found here. The authors also have their own paper discussing how well the Heston-Model performs with respect to empirical stock returns (Comparison between the probability distribution
of returns in the Heston model and empirical
data for stock indexes)
I can also wholeheartedly recommend the paper on the Goodness-of-fit of the Heston, Variance-Gamma and
Normal-Inverse Gaussian Models
It is very extensive, comprehensive and simultaneously provides a good introduction to the respective models
Due to it's fat tails the student-t distribution also performes reasonably well at reproducing market stock returns. As Aksakal has already mentioned in the comments below Student t is not a stable distribution. Still, there are some references using it to fit stock returns and even to construct a random walk. For a reference on the prior confer the paper by Eckhard Platen on Empirical Evidence
on Student-tLog-Returns of Diversied World Stock Indices. A construction of student t random walk can be found in A Benchmark Approach to Quantitative Finance by Platen and Heath.
How to go from transition density to the distribution of returns
My explanation above did not calrify how knowing the transition density of a process also gives information on the distribution/density of the returns.
Assume a process $\ln(X_t)$ has the transition density $p(s,x,t,y)$. $p(s,x,t,y)$ can be interpreted as the probability of the process to move from $X_s=x$ at time $s$ to $X_t=y$ at time $t$. Thus if we set $s=0, x=0$ and $t=\Delta t$ (e.g. one day, one weel etc) we arrive at a density $p(0,0,\Delta t,y)$. This function describes the probability of the process to increass by $y$ within a time step of $\Delta t$. Thus depending on choice of $\Delta t$ wone is able to retrieve the theoretical. distribution of daily, montly, etc. returns.