A double Roman dominating function on a graph G is a function f:VG⟶0,1,2,3 satisfying the conditions that every vertex u for which fu=0 is adjacent to at least one vertex v for which fv=3 or two vertices v1 and v2 for which fv1=fv2=2 and every vertex u for which fu=1 is adjacent to at least one vertex v for which fv≥2. The weight of a double Roman dominating function f is the value fV=∑u∈Vfu. The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination numberγdRG of G. A graph with γdRG=3γG 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 γdRT=γRT+k for k=1,2.