The development of data-informed predictive models for dynamical systems is of widespread interest in many disciplines.We present a unifying framework for blending mechanistic and machine-learning approachesto identify dynamical systems from noisily and partially observed time-series data.Our formulation is agnostic to the chosen machine learning model,is presented in both continuous- and discrete-time settings,and is compatible both with systems that exhibit substantial memory and systems that are memoryless.We conclude with a series of numerical results thata) illustrate trade-offs when learning dynamics in continuous- and discrete-time,and b) demonstrate the inference power of our methodology in a partially observed Lorenz '63 system.