Although the other answers have already addressed the question, I would like to add another powerful option that would solve most of the problems related to the distribution-assumptions: quantile regression.
Depending on the research interests, this method can be extremely powerful.
As someone has already said before, if you are merely interested in estimating the marginal mean (or any quantile) of your outcome then you don't need to worry about any assumption at all, as both quantile and ordinary regression methods perfectly estimate it.
If you are interested in inference, ordinary regression has a couple of problems with the distribution assumptions, whereas the quantile doesn't because it is distribution free.
It's true that you can try using mean regression and robust estimators, but personally I prefer quantile regression, which is by the way even more informative (because you can estimate the whole conditional distribution of the outcome instead of just one of its summary indicators, the mean).
If you are interested in both prediction and inference, then the quantile's property of invariance is quite handy.
For example, suppose you are working with probabilities, or rates (or any other "bounded" outcome).
With quantile regression you can transform the outcome Y so that it's transformation is not bounded (for example, using a logit or probit function), model logit(Y) and use the same model for predictions and inference.
With ordinary methods it's not so easy, because of Jensen's inequality: E(g(Y)) is never equal to g(E(Y)).
Therefore, you either use two models (one for the prediction, one for the association) or you must use other methods (beta regression, logit normal regression) that, however, have problems related to respectively parameter interpretation and distribution assumptions.
Finally, there can always be problems related to the linearity assumption or independent data. In the former, we can solve the problem by adding splines (which, though, complicate the interpretation of parameters).
For the latter, instead, mixed effect regression models could help us (if we have hierarchical or longitudinal data).