  Domination game [SIAM J.\ Discrete Math.\ 24 (2010) 979--991] and total
domination game [Graphs Combin.\ 31 (2015) 1453--1462] are by now well
established games played on graphs by two players, named Dominator and Staller.
In this paper, Z-domination game, L-domination game, and LL-domination game are
introduced as natural companions of the standard domination games.
  Versions of the Continuation Principle are proved for the new games. It is
proved that in each of these games the outcome of the game, which is a
corresponding graph invariant, differs by at most one depending whether
Dominator or Staller starts the game. The hierarchy of the five domination
games is established. The invariants are also bounded with respect to the
(total) domination number and to the order of a graph. Values of the three new
invariants are determined for paths up to a small constant independent from the
length of a path. Several open problems and a conjecture are listed. The latter
asserts that the L-domination game number is not greater than $6/7$ of the
order of a graph.
