  The primary sample of the {\it Gaia} Data Release 1 is the Tycho-Gaia
Astrometric Solution (TGAS): $\approx$ 2 million Tycho-2 sources with improved
parallaxes and proper motions relative to the initial catalog. This increased
astrometric precision presents an opportunity to find new binary stars and
moving groups. We search for high-confidence comoving pairs of stars in TGAS by
identifying pairs of stars consistent with having the same 3D velocity using a
marginalized likelihood ratio test to discriminate candidate comoving pairs
from the field population. Although we perform some visualizations using (bias-
corrected) inverse parallax as a point estimate of distance, the likelihood
ratio is computed with a probabilistic model that includes the covariances of
parallax and proper motions and marginalizes the (unknown) true distances and
3D velocities of the stars. We find 13,085 comoving star pairs among 10,606
unique stars with separations as large as 10 pc (our search limit). Some of
these pairs form larger groups through mutual comoving neighbors: many of these
pair networks correspond to known open clusters and OB associations, but we
also report the discovery of several new comoving groups. Most surprisingly, we
find a large number of very wide ($>1$ pc) separation comoving star pairs, the
number of which increases with increasing separation and cannot be explained
purely by false-positive contamination. Our key result is a catalog of
high-confidence comoving pairs of stars in TGAS. We discuss the utility of this
catalog for making dynamical inferences about the Galaxy, testing stellar
atmosphere models, and validating chemical abundance measurements.
