This paper investigates representation of a multivariable linear time-invariant system by matrix fraction descriptions (MFD). The main theoretical results in our recent paper [1] have resolved several long-standing issues of MFD models, including uniqueness of MFD pairs fA(z) B(z)g for a given impulse response of the system, complete characterization of the orders of all possible MFDs of a given system, testing criteria for determining whether a matrix pair is an MFD of the system, and algorithms for deriving all MFD representations from certain Toeplitz matrices constructed from impulse responses. For real-time implementation, identification algorithms are introduced that estimate all MFDs of a given system from its input-output data. The results are then extended to cover ARMAX systems. This conference paper aims to reach a broader audience and stimulate potential applications of the methodology and algorithms by highlighting the main results, elaborating key ideas with more discussions, extending simulation studies, and including more examples.