My aim is to make some comments of a quite general nature about the relation between applied mathematics and finance, theoretical and practical. I shall begin with a brief description of a case in which ‘technology transfer’ from a quite different area of mathematics, the Stefan problem, was helpful with a financial problem, namely the Black-Scholes approach to an American option. I then discuss some more general issues about the role of this kind of mathematics in finance and suggest some possible avenues for future progress.