A modular form is a certain holomorphic function on the complex upper half plane which has an infinite group of discrete symmetries. I will discuss modular forms and some of their generalizations.  In particular, I will try to answer the following questions: What is a modular form? What are some examples of modular forms? What is a simple open question about modular forms?  The examples of modular forms I will give go under the name of "harmonic theta functions".  Time permitting, I will describe some exotic variants of these harmonic theta functions that are tied up with the octonions and the compact Lie group G_2.