Traditional time series models assume discrete, evenly-spaced observations, yet real-world data—from irregularly sampled electronic health records to high-frequency financial ticks—exhibit continuous temporal dynamics with irregular sampling, missing values, and non-stationary regimes. This fundamental mismatch between discrete model assumptions and continuous data reality creates systematic bias and limits predictive accuracy. This chapter introduces continuous time series analysis as a paradigm shift that treats observations as realizations of underlying continuous stochastic processes governed by differential operators. The core framework centers on Neural Ordinary Differential Equations (Neural ODEs) and Neural Stochastic Differential Equations (Neural SDEs), which embed continuous evolution directly into learning architectures, enabling adaptive temporal resolution and causally consistent inference. The unifying insight is that separating dynamics from sampling—by modeling the continuous process Z(t) and treating observations as restrictions to random time sets—eliminates discretization bias and naturally handles irregularity, missingness, and variable latency. This mathematically coherent foundation enables robust reasoning over complex temporal systems spanning physiological processes, climate patterns, and financial volatility, where time flows continuously rather than in fixed steps, establishing the theoretical groundwork for all subsequent chapters on continuous-time neural architectures.