Energy harvesting (EH) based communication has raised great research interests due to its wide application and the feasibility of commercialization. In this paper, we consider wireless communications with EH constraints at the transmitter. First, for delay-tolerant traffic, we investigate the long-term average throughput maximization problem and analytically compare the throughput performance against that of a system supported by conventional power supplies. Second, for delay-sensitive traffic, we analyze the outage probability by studying its asymptotic behavior in the high energy arrival rate regime, where the new concept of energy diversity is formally introduced. Moreover, we show that the speed of outage probability approaching zero, termed energy diversity gain, varies under different power supply models.