This dissertation investigates some topics involving periodic autoregressive moving- average (PARMA) time series models.|Our first research topic studies autocovariance and partial autocorrelation properties of PARMA models. An efficient algorithm to compute PARMA autocovariances is first derived. An Innovations based algorithm to compute partial autocorrelations for a general periodic series is then developed. Periodic moving-averages and periodic autoregressions are characterized as periodically stationary series whose autocovariances and partial autocorrelations, respectively, are zero at all lags that exceed some periodically varying threshold.|Next, techniques for fitting parsimonious periodic time series models are explored. Large sample standard errors for the parameter estimates of a PARMA model under parametric constraints are derived; likelihood ratio statistics are also explored. The techniques are motivated with the analysis of a daily temperature series from Griffin, Georgia.|The dissertation closes by introducing seasonal periodic autoregressive moving- average time series (SPARMA) models. SPARMA models are a hybrid of seasonal autoregressive moving-average models and PARMA models. Some mathematical properties of SPARMA models are derived.