Inverse problems integrate computational simulations of natural phenomena with physical measurements in an information feedback control system. Control parameters of the computational model are optimized to fit the physical measurements. Inverse problems arise in many scientific and engineering applications such as weather prediction and geophysical exploration. Typically, the solution procedure is computationally expensive -- involving the simulation of high-dimensional forward and the corresponding adjoint model. In practice, our knowledge of the underlying physics is incomplete and hence the associated forward model is laden with model errors. Similarly, it is not possible to measure the physical quantities exactly and hence the measurements are associated with data errors. Additionally, malfunctioning sensors lead to incorrect measurements, causing outliers in the data. This presentation discusses a computational machinery to measure the impact of model and data errors on the solution and a robust optimization framework that is insensitive to data outliers. The talk will also present dimension-reduction strategies and a parallel-in-time algorithm to mitigate the high computational costs.