A generalized framework for single-variate and multivariate simulation and forecasting problems in stochastic hydrology is proposed. It is appropriate for short-term or long-term memory processes and preserves the Hurst coefficient even in multivariate processes with a different Hurst coefficient in each location. Simultaneously, it explicitly preserves the coefficients of skewness of the processes. The proposed framework incorporates short memory (autoregressive - moving average) and long memory (fractional Gaussian noise) models, considering them as special instances of a parametrically defined generalized autocovariance function, more comprehensive than those used in these classes of models. The generalized autocovariance function is then implemented in a generalized moving average generating scheme that yields a new time symmetric (backward-forward) representation, whose advantages are studied. Fast algorithms for computation of internal parameters of the generating scheme are developed, appropriate for problems including even thousands of such parameters. The proposed generating scheme is also adapted through a generalized methodology to perform in forecast mode, in addition to simulation mode. Finally, a specific form of the model for problems where the autocorrelation function can be defined only for a certain finite number of lags is also studied. Several illustrations are included to clarify the features and the performance of the components of the proposed framework.