  The cohomology theory known as Tmf, for "topological modular forms," is a
universal object mapping out to elliptic cohomology theories, and its
coefficient ring is closely connected to the classical ring of modular forms.
We extend this to a functorial family of objects corresponding to elliptic
curves with level structure and modular forms on them. Along the way, we
produce a natural way to restrict to the cusps, providing multiplicative maps
from Tmf with level structure to forms of K-theory. In particular, this allows
us to construct a connective spectrum tmf_0(3) consistent with properties
suggested by Mahowald and Rezk.
  This is accomplished using the machinery of logarithmic structures. We
construct a sheaf of locally even-periodic elliptic cohomology theories,
equipped with highly structured multiplication, on the log-\'etale site of the
moduli of elliptic curves. Evaluating this sheaf on modular curves produces Tmf
with level structure.
