Coupled-inductor interleaved boost converters are under development for high-current, high-power applications ranging from automotive to distributed generation. The operating modes of these coupled-inductor converters can be complex. This paper presents an investigation of the various continuous-current (CCM) and discontinuous-current (DCM) modes of operation of the coupled-inductor interleaved two-phase boost converter. The various CCM and DCM of the converter are identified together with their submodes of operation. The standard discrete-inductor interleaved two-phase boost can be seen as a subset of the coupled-inductor converter family with zero mutual coupling between the phases. The steady-state operating characteristics, equations and waveforms for the many CCM and DCM will be presented for the converter family. Mode maps will be developed to map the converter operation across the modes over the operating range. Experimental validation is presented from a 3.6 kW laboratory prototype. Design considerations and experimental results are presented for a 72 kW prototype.