A double Roman dominating function on a graph G is a function f : V G ⟶ 0,1,2,3 satisfying the conditions that every vertex u for which f u = 0 is adjacent to at least one vertex v for which f v = 3 or two vertices v 1 and v 2 for which f v 1 = f v 2 = 2 and every vertex u for which f u = 1 is adjacent to at least one vertex v for which f v ≥ 2 . The weight of a double Roman dominating function f is the value f V = ∑ u ∈ V f u . The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination number γ d R G of G . A graph with γ d R G = 3 γ G is called a double Roman graph. In this paper, we study properties of double Roman domination in graphs. Moreover, we find a class of double Roman graphs and give characterizations of trees with γ d R T = γ R T + k for k = 1,2 .