The brushless doubly-fed machine(BDFM) is a family of multiport electric machines with two ac electrical ports and a common mechanical port. Different from the conventional singly-fed machines whose synchronous speed is solely determined by a single supply frequency and the actual pole pair number, the BDFM has two supply frequencies and two different pole pair numbers to control the rotor speed. By the two accessible electrical ports, all BDFMs are endowed with more degrees of freedom for speed and power control, inherent fault-tolerant capability and high reliability. The BDFM in its broad sense has been extensively investigated as a promising alternative to the conventional slip-ring doubly-fed induction machine(DFIM) during the past decades, for both limited and wide speed range applications. This paper presents a new theoretical framework of the BDFM within which all topological variants can be closely linked by the similarities in working principle. The individualities of each machine topology are presented first, followed by the commonalities such as the modeling techniques, modes of operation, design considerations and control strategies. The challenges are identified and highlighted based on recent developments and possible opportunities are predicted considering the unique nature of this special AC machine type.