Abel's theorem (1824) that the generic polynomial of degree n is solvable in radicals if and only if $n$ $\<$ 5 is well-known. However, the classical works of Bring (1786) and Klein (1884) give solutions of the generic quintic polynomial by allowing certain other ``nice'' algebraic functions of one variable. For the sextic, it is conjectured that any solution requires algebraic functions of two variables. In this talk, we will examine and relate the intrinsic geometries of the known solutions of the sextic in two variables, extending the work of Green (1978).