  We study a suite of extremely high-resolution cosmological FIRE simulations
of dwarf galaxies ($M_{\rm halo} \lesssim 10^{10}$$M_{\odot}$), run to $z=0$
with $30 M_{\odot}$ resolution, sufficient (for the first time) to resolve the
internal structure of individual supernovae remnants within the cooling radius.
Every halo with $M_{\rm halo} \gtrsim 10^{8.6} M_{\odot}$ is populated by a
resolved {\em stellar} galaxy, suggesting very low-mass dwarfs may be
ubiquitous in the field. Our ultra-faint dwarfs (UFDs;
$M_{\ast}<10^{5}\,M_{\odot}$) have their star formation truncated early
($z\gtrsim2$), likely by reionization, while classical dwarfs ($M_{\ast}>10^{5}
M_{\odot}$) continue forming stars to $z<0.5$. The systems have bursty star
formation (SF) histories, forming most of their stars in periods of elevated SF
strongly clustered in both space and time. This allows our dwarf with
$M_{\ast}/M_{\rm halo} > 10^{-4}$ to form a dark matter core $>200$pc, while
lower-mass UFDs exhibit cusps down to $\lesssim100$pc, as expected from
energetic arguments. Our dwarfs with $M_{\ast}>10^{4}\,M_{\odot}$ have
half-mass radii ($R_{\rm 1/2}$) in agreement with Local Group (LG) dwarfs;
dynamical mass vs. $R_{1/2}$ and the degree of rotational support also resemble
observations. The lowest-mass UFDs are below surface brightness limits of
current surveys but are potentially visible in next-generation surveys (e.g.
LSST). The stellar metallicities are lower than in LG dwarfs; this may reflect
pre-enrichment of the LG by the massive hosts or Pop-III stars. Consistency
with lower resolution studies implies that our simulations are numerically
robust (for a given physical model).
