Is there any general method of solving polynomials of arbitrary degree $n$? By ''solving'' I mean expressing roots of polynomial in terms of some reasonable special functions and coefficients in a computable manner.

I know that 'reasonable' is extremely vague here. I mean functions which has been investigated in some other context or ones which fit into 'nice' families governed by parameters from $\mathbb{N}$, say, as solutions to a family of ODEs.

I suppose that the answer should be 'no', since otherwise no one would care about solutions to the polynomials of degree 5 with elliptic functions.