  In the past decades, advanced probabilistic methods have had significant
impact on the field of finance, both in academia and in the financial industry.
Conversely, financial questions have stimulated new research directions in
probability. In this survey paper, we review some of these developments and
point to some areas that might deserve further investigation. We start by
reviewing the basics of arbitrage pricing theory, with special emphasis on
incomplete markets and on the different roles played by the "real-world"
probability measure and its equivalent martingale measures. We then focus on
the issue of model ambiguity, also called Knightian uncertainty. We present two
case studies in which it is possible to deal with Knightian uncertainty in
mathematical terms. The first case study concerns the hedging of derivatives,
such as variance swaps, in a strictly pathwise sense. The second one deals with
capital requirements and preferences specified by convex and coherent risk
measures. In the final two sections we discuss mathematical issues arising from
the dramatic increase of algorithmic trading in modern financial markets.
