  We study fixed-points of scalar fields that transform in the bifundamental
representation of $O(N)\times O(M)$ in $3-\epsilon$ dimensions, generalizing
the classic tricritical sextic vector model. In the limit where $N$ is large
but $M$ is finite, we determine the complete beta function to order $1/N$ for
arbitrary $M$. We find a rich collection of large-$N$ fixed-points in $d=3$, as
well as fixed-points in $d=3-\epsilon$, that can be studied to all orders in
the parameter $\hat{\epsilon}=N\epsilon$. With the goal of defining a large-$N$
nonsupersymmetric conformal field theory dominated by a web of planar diagrams,
we also study fixed-points in the ``bifundamental'' large-$N$ limit, in which
$M$ and $N$ are both large, but the ratio $M/N$ is held fixed. We find a unique
infrared fixed-point in $d=3-\epsilon$, which we determine to order
$\epsilon^2$. When $M/N \ll 1$, we also find an ultraviolet fixed-point in
$d=3$ and $d=3-\epsilon$ that merges with the infrared fixed-point at $\epsilon
\sim O(M/N)$. We expect at least one of two candidate fixed-points in integer
dimensions -- the infrared fixed-point in $d=2$ and the ultraviolet fixed-point
in $d=3$ -- to survive for finite values of $M/N$.
