Forecasting the evolution of complex systems is noted as one of the 10 grand challenges of modern science. Time series data from complex systems capture the dynamic behaviors and causalities of the underlying processes and provide a tractable means to predict and monitor system state evolution. However, the nonlinear and non-stationary dynamics of the underlying processes pose a major challenge for accurate forecasting. For most real-world systems, the vector field of state dynamics is a nonlinear function of the state variables; i.e., the relationship connecting intrinsic state variables with their autoregressive terms and exogenous variables is nonlinear. Time series emerging from such complex systems exhibit aperiodic (chaotic) patterns even under steady state. Also, since real-world systems often evolve under transient conditions, the signals obtained therefrom tend to exhibit myriad forms of non-stationarity. Nonetheless, methods reported in the literature focus mostly on forecasting linear and stationary processes. This article presents a review of these advancements in nonlinear and non-stationary time series forecasting models and a comparison of their performances in certain real-world manufacturing and health informatics applications. Conventional approaches do not adequately capture the system evolution (from the standpoint of forecasting accuracy, computational effort, and sensitivity to quantity and quality of a priori information) in these applications.