  A theta graph, denoted $\theta_{a,b,c}$, is a graph of order $a+b+c-1$
consisting of a pair of vertices and three independent paths between them of
lengths $a$, $b$, and $c$. We provide a complete characterization of graphs
that do not contain a large $\theta_{a,b,c}$ as a topological minor. More
specifically, we describe the structure of $\theta_{1,2,t}$-,
$\theta_{2,2,t}$-, $\theta_{1,t,t}$-, $\theta_{2,t,t}$-, and
$\theta_{t,t,t}$-free graphs where $t$ is large. The main result is a
characterization of $\theta_{t,t,t}$-free graphs for large $t$. The
$3$-connected $\theta_{t,t,t}$-free graphs are formed by $3$-summing graphs
without a long path to certain planar graphs. The $2$-connected
$\theta_{t,t,t}$-free graphs are then built up in a similar fashion by 2- and
3-sums. This result implies a well-known theorem of Robertson and Chakravarti
on graphs that do not have a bond containing three specified edges.
