The notion of an attractor for a random dynamical system with respect to a general collection of deterministic sets is introduced. This comprises, in particular, global point attractors and global set attractors. After deriving a necessary and sufficient condition for existence of the corresponding attractors it is proved that a global set attractor always contains all unstable sets of all of its subsets. Then it is shown that in general random point attractors, in contrast to deterministic point attractors, do not support all invariant measures of the system. However, for white noise systems it holds that the minimal point attractor supports all invariant Markov measures of the system.