Mathematical biology is a huge area which is not so young.
Statistics is a major research tool in biology (as in most other areas of natural and social science) so biology questions rely and have led to substantial progress in mathematical statistics and related probability theory.
Like in most natural sciences differential equations of various types (and some unique types of euations) arose in biology. It makes sense to mention in particular the pioneering work of Alan Turing in his paper entitled The Chemical Basis of Morphogenesis. Nilima Nigam's answer mentioned several related connections and this is just the tip of the iceberg.
Mathematical biology have led to Evolutionary game theory which had strong impact on mathematical game theory.
Sometimes a single question in biology is related to a large number of mathematical disciplines. For example, Amit Singer and Yoel Shkolnisky are involved in a long term project aiming to determine the 3-dimensional shape of a certain molecule based on noisy 2-dimensional pictures from an electronic microscope. Their research is related to fascinating questions in harmonic analysis (and wavelets), graph theory, representation theory, semidefinite programmings, probability theory, and even the notion of unique games fron theoretical computer science enter.
A last example: John Bush and David Hu had remarlable mathematical models explaining how insects walk on waters. Bush's homepage is a good source for various issues in mathematical biology.