Continuous-time time series pose structural challenges—irregular sampling, latent diffusion, and non-Markovian dependencies—that demand models capable of representing both dynamics and uncertainty. This chapter surveys the probabilistic foundations underlying such systems, beginning with classical approaches rooted in stochastic differential equations, Gaussian processes, and state—space models. We then examine deep probabilistic architectures, where variational inference, normalizing flows, and diffusion-based generative models approximate high-dimensional temporal likelihoods through latent stochastic processes. Generative frameworks unify forecasting and simulation by modeling the full trajectory distribution rather than point estimates, while Bayesian neural networks and ensemble methods propagate epistemic uncertainty under data sparsity. Through a comparative analysis, we highlight the computational trade-offs, expressivity limits, and calibration properties of competing paradigms. The chapter concludes with open research challenges—including identifiability, stability of continuous-time inference, and the design of priors that respect temporal structure—thereby outlining a path toward principled and uncertainty-aware forecasting in continuous-time domains.