The energy of a graph G, E (G), is the sum of absolute values of the eigenvalues of its adjacency matrix. Gutman et al. proved that for every cubic graph of order n, E (G)≥ n. Here, we improve this result by showing that if G is a connected subcubic graph of order n≥ 8, then E (G)≥ 1.01 n. Also, we prove that if G is a traceable subcubic graph of order n≥ 8, then E (G)> 1.1 n. Let G be a connected cubic graph of order n≥ 8, it is shown that E (G)> n+ 2. It was proved that if G is a connected cubic graph of order n, then E (G)≤ 1.65 n. Also, in this paper we would like to present the best lower bound for the energy of a connected cubic graph. We introduce an infinite family of connected cubic graphs whose for each element of order n, say G, E (G)≥ 1.24 n, and conjecture that if 6| n, then minimum energy occurs just for each element of this family. We conjecture that there exists N such that for every connected cubic graph G of order n≥ N, E (G)≥ 1.24 n.