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This review focuses on the use of Bayesian Data Analysis and Machine Learning Techniques to study and analyze flow problems typical to polymer melt systems. We present a brief summary of Bayesian probability theory, and show how it can be used to solve the parameter estimation and model selection problems, for cases when the model(s) are known. For the more complex non-parametric regression problem, in which the functional form of the model is not known, we show how Machine-Learning (through Gaussian Processes) can be used to learn arbitrary functions from data. In particular, we show examples for solving steady-state flow problem as well as learning the constitutive relations of polymer flows with memory.