To every sufficiently nice vertex operator algebra (VOA), one can assign a number of modular forms. First, we define (in part) vertex operator algebras, introducing the Heisenberg VOA and the VOAs constructed from even self-dual lattices. We then consider elements of the lattice VOAs defined using harmonic polynomials, and compute the associated modular forms. Finally, we look at the analogous elements of more general VOAs and say something about their associated modular forms.