Mirror symmetry was a phenomenon discovered by physicists
around 1989: they observed that certain kinds of six-dimensional
geometric objects known as Calabi-Yau manifolds seemed to come in
pairs, with a strange relationship between different kinds of
geometric objects on the pairs. Since then, the subject has blossomed
into a vast field, with many different approaches and philosophies. I
will give a brief introduction to the subject, and explain how one of
these approaches, developed with Bernd Siebert, has led to a general
construction of mirror pairs.