A graph G = (V, E) is called super edge-magic if there exists a one-to-one map $lambda$ from V $cup$ E onto {1,2,3,...,|V|+|E|} such that $lambda$(V)={1,2,...,|V|} and $lambda(x)+lambda(xy)+lambda(y)$ is constant for every edge xy. In this paper, we investigate whether some families of graphs are super edge-magic or not.