Robust statistics answers the question of how to build statistical estimators that behave well even when a small fraction of the input data is badly corrupted. While the information-theoretic underpinnings have been understood for decades, until recently all reasonably accurate estimators in high dimensions were computationally intractable. Recently however, a new class of algorithms has arisen that overcome these difficulties providing efficient and nearly-optimal estimates. Furthermore, many of these techniques can be adapted to cover the case where the majority of the data has been corrupted. These algorithms have surprising applications to clustering problems even in the case where there are no errors.