The existence of a maximal solution to some typical fuzzy relational equations is proved in the paper, in case when the codomain lattice is complete and meet-continuous, which means that the infimum commutes with the supremum of chains. These results extend the existing results, which are mostly limited to the existence of a maximal solution to some typical fuzzy set and fuzzy relational inequations. In order to prove that the same holds for the corresponding relational equations, it is proved that the property of meet-continuity in the codomain lattice implies another property in the lattice of fuzzy relations, namely that the composition of fuzzy relations, defined as usual, commutes with the supremum of chains. This condition can also be taken instead of the meet-continuity of the codomain lattice as another sufficient condition for the existence of a maximal solution to the equations considered here. Two examples are given, which prove that these conditions do not imply the existence of the greatest solution to some of the considered equations.