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A wide range of parameters of domination in graphs can be defined and studied through a common approach that was recently introduced in [] under the name of w-domination, where w = (w 0 , w 1 , ⋯ , w l) is a vector of non-negative integers such that w 0 ≥ 1 . Given a graph G, a function f : V (G) ⟶ { 0 , 1 , ⋯ , l } is said to be a w-dominating function if ∑ u ∈ N (v) f (u) ≥ w i for every vertex v with f (v) = i , where N(v) denotes the open neighbourhood of v ∈ V (G) . The weight of f is defined to be ω (f) = ∑ v ∈ V (G) f (v) , while the w-domination number of G, denoted by γ w (G) , is defined as the minimum weight among all w-dominating functions on G. A wide range of well-known domination parameters can be defined and studied through this approach. For instance, among others, the vector w = (1 , 0) corresponds to the case of standard domination, w = (2 , 1) corresponds to double domination, w = (2 , 0 , 0) corresponds to Italian domination, w = (2 , 0 , 1) corresponds to quasi-total Italian domination, w = (2 , 1 , 1) corresponds to total Italian domination, w = (2 , 2 , 2) corresponds to total { 2 } -domination, while w = (k , k - 1 , ⋯ , 1 , 0) corresponds to { k } -domination. In this paper, we show that several domination parameters of lexicographic product graphs G ∘ H are equal to γ w (G) for some vector w ∈ { 2 } × { 0 , 1 , 2 } l and l ∈ { 2 , 3 } . The decision on whether the equality holds for a specific vector w will depend on the value of some domination parameters of H. In particular, we focus on quasi-total Italian domination, total Italian domination, 2-domination, double domination, total { 2 } -domination, and double total domination of lexicographic product graphs. [ABSTRACT FROM AUTHOR]