We study the channel capacity of a general discrete energy harvesting channel with a finite battery. Contrary to traditional communication systems, the transmitter of such a channel is powered by a device that harvests energy from a random exogenous energy source and has a finite-sized battery. As a consequence, at each transmission opportunity the system can only transmit a symbol whose energy is no more than the energy currently available. This new type of power supply introduces an unprecedented input constraint for the channel, which is simultaneously random, instantaneous, and influenced by the full history of the inputs and the energy harvesting process. Furthermore, naturally, in such a channel the energy information is observed causally at the transmitter. Both of these characteristics pose great challenges for the analysis of the channel capacity. In this work we use techniques developed for channels with side information and finite state channels, to obtain lower and upper bounds on the capacity of energy harvesting channels. In particular, in a general case with Markov energy harvesting processes we use stationarity and ergodicity theory to compute and optimize the achievable rates for the channels, and derive series of computable capacity upper and lower bounds.