Discrete-time linear periodically time-varying (LPTV) systems, considered as a bridge between the well-studied linear time-invariant (LTI) model and the nonlinear time-varying problems in real world, have been receiving increasing attention in recent a few decades. In this research project, we try to understand discrete-time LPTV systems both internally and externally and derive basic theories for analysis, realization and model reduction of LPTV systems. Firstly we review the system model for LPTV systems, define its transfer function matrix, Markov parameters, stability, reachability and observability. Then we emphasize on the numerically efficient and stable methods to compute LPTV system grammians and to approximate the eigenvalue decay rate. Another main result of this thesis is Krylov-based moment matching algorithm for model reduction of LPTV systems, which is derived afterwards, and is also compared to the other approach: balancing and balanced truncation of LPTV systems. Almost any application of discrete-time LPTV systems, including periodic digital filters and periodic control theories, demands a periodic state-space model from input-output maps. This periodic realization problem is treated at the end of the thesis with demonstration of applicable non-minimal and quasi-minimal realization methods.