For an arbitrary subset P of the reals, a function f: V → P is defined to be a P-dominating function of a graph G = (V, E) if the sum of its function values over any closed neighbourhood is at least 1. That is, for every v ∈ V,f(N[v]) ≥ 1. The definition of total P-dominating function is obtained by simply changing closed neighborhood N[v] in the definition of P-dominating function to open neighborhood N(v). The (total) P-domination number of a graph G is defined to be the infimum of weight w(f) = Σv∈V f(v) taken over all (total)P-dominating function f. Similarly, the P-edge and P-star dominating functions can be defined. In this paper we survey some recent progress on the topic of dominating functions in graph theory. Especially, we are interested in P-, P-edge and P-star dominating functions of graphs with integer values.