Incomplete knowledge of the true dynamics and its partial observations pose a notoriously difficult problem in many contemporary scientific applications which require predictions of high-dimensional dynamical systems with physical instabilities and energy fluxes across a wide range of scales. The issue of `model error' is particularly important when dealing with turbulent geophysical systems or molecular dynamics with rough energy spectra near the resolution cut-off of the numerical models. In such cases assimilation of observed data into the modeled dynamics is necessary for mitigating model error and for improving the stability and predictive skill of the imperfect models. In the talk I will give an overview of my research on a newly emerging stochastic-statistical framework which allows for information-theoretic quantification of uncertainty and mitigation of model error in imperfect statistical predictions of complex multi-scale dynamics. Two important examples used to highlight these issues will be concerned with (i) existence of `information' barriers to imperfect model improvement and (ii) real-time `stochastic superresolution' for estimation of the relevant unresolved processes in sparsely observed turbulent systems. Open mathematical and practical problems for future research will also be discussed.