The conformal method is historically the most studied method for (trying to) parameterize the set of solutions to the Einstein constraint equations. While it gives satisfactory results in the constant mean curvature (CMC) case, its application to the non-CMC case appears extremely difficult. Both methods known so far to solve the conformal constraint equations are blind in one eye. Each sees some part of the problem but not the global structure. In this talk I will discuss some recent advances on each method.