  We study the renormalization group flow of $\phi^4$ theory in two dimensions.
Regularizing space into a fine-grained lattice and discretizing the scalar
field in a controlled way, we rewrite the partition function of the theory as a
tensor network. Combining local truncations and a standard coarse-graining
scheme, we obtain the renormalization group flow of the theory as a map in a
space of tensors. Aside from qualitative insights, we verify the scaling
dimensions at criticality and extrapolate the critical coupling constant
$f_{\rm c} = \lambda / \mu ^2$ to the continuum to find $f^{\rm cont.}_{\rm c}
= 11.0861(90)$, which favorably compares with alternative methods.
