Taking into account the effect of the low-energy band structure of carbon nanotubes (CN), we develop the theories of CN field and thermionic emissions. We give the analytic field and thermionic emission equations for both metal and semiconducting CNs. These theories modify the conventional Fowler-Nordheim (FN) and Murphy-Good (MG) theories. For large-diameter CNs and high fields, the field-emission equation reduces the FN-type field-emission equation. For small-diameter CNs and low fields, the field-emission equation goes beyond the FN-type behavior, which provides a possible way to understand the non-FN behavior observed in experimental results. Based on these theories, we give the electron-emission phase diagram on the field, thermionic, and intermediate emissions in the field-temperature space, whose boundaries have a slight shift to the corresponding boundaries of MG's theory. These differences come from the energy-band structure difference between CN and conventional emitters. This theory provides an understanding of field and thermionic emissions for nanoscale materials.