Living systems gather and transmit information about the internal and external
environments through biochemical networks of interacting molecules. The robustness
and fidelity of information processing in biochemical networks can be limited by
noise, the structure of biochemical interactions or numbers of participating
molecules. Living cells, as a whole, are highly dissipative and it has been long
recognized that information processing too require consumption of energy. Less is
known about the limits and energetic cost of the capacity of information processing
through biologically relevant biochemical reaction networks. Combining
non-equilibrium statistical physics and learning theory perspectives we derive
fundamental constraints relating  the amount of dissipated energy and the fidelity
and speed of information processing in individual cell signaling modules. We also
derive the limits on equilibrium modules imposed by the available complexity and
cooperativity but also by the range of input signals. Finally we analyze the
information processing requirements to the energy budget of the cell.