A Probability Metrics Approach to Financial Risk Measures relates the field of probability metrics and risk measures to one another and applies them to finance for the first time. * Helps to answer the question: which risk measure is best for a given problem? * Finds new relations between existing classes of risk measures * Describes applications in finance and extends them where possible * Presents the theory of probability metrics in a more accessible form which would be appropriate for non-specialists in the field * Applications include optimal portfolio choice, risk theory, and numerical methods in finance * Topics requiring more mathematical rigor and detail