This dissertation addresses trends in the death of COVID-19 in Hungary and energy consumption patterns in Brazil, with the objective of modeling and forecasting time-series data using sophisticated statistical methodologies. Initially, a unified theoretical framework was established by deriving the state space representations for core time series models (AR, MA, ARMA, ARIMA, HW). Secondly, we utilized World Health Organization data from 2020 to 2021 and employed a Seasonal ARIMA model to forecast daily death from COVID-19 in Hungary. The fitted SARIMA model (1,1,2)(1,0,1)[7] indicates that the residuals were normally distributed and exhibited significant metrics. The daily fatalities were anticipated to decrease. The subsequent phase of study findings involved assessing the potential enhancement of the Double Seasonal Holt-Winters model through the incorporation of ARMA(3,1) errors. For energy consumption data with different seasonalities, in particular, the combined (modified) model improved prediction reliability and successfully addressed residual autocorrelation, outperforming standard DSHW across various performance metrics (ME, RMSE, MAPE). Lastly, a modified method based on periodograms was implemented to identify various seasonalities (daily, half-daily and sub-daily) in data on energy consumption collected hourly. In the analysis of multi-seasonal trends, STL + ETS(A,N,N) exhibited the most accurate forecasts compared to the BATS and TBATS models. The dissertation emphasizes the need of choosing appropriate time series models for various seasonal structures, giving practitioners significant insights for decision-making in dynamic situations. Key contributions include formulating the state space representations for core time series models, enhanced SARIMA forecasting for epidemiological data, better DSHW-ARMA combined modeling, and a strong framework for selecting dominant frequency and modeling multi-seasonal time series.