<p>Cylindrical graphs and torus grid graphs are naturally constructed from subgraphs of the infinite grid by certain identifications of boundary vertices. Considering various domination type problems, it is usually possible to find an optimal solution on the infinite grid. To the contrary, exact values of invariants for the cylindrical and torus grid graphs are typically only known for special subfamilies, and are in general hard to compute. The 2-domination and 2-rainbow domination of cylindrical graphs is studied, and some new formulae and improved bounds are reported. We also consider weak 2-domination and singleton rainbow domination.</p>