This lecture is part of an online graduate course on modular forms. We introduce modular forms, and give several examples of how they were used to solve problems in apparently unrelated areas of mathematics. I will not be following any particular book, but if anyone wants a suggestion for a book I recommend "A course on arithmetic" by . Serre (Chapter VII) and "Modular functions and Dirichlet series in number theory" by T. Apostol. Typo: at 7:05 the definition of q at the bottom is missing a minus sign in the exponent. For the other lectures in the course see