  We review the calculation of the Hoyle state of $^{12}$C in Nuclear Lattice
Effective Field Theory (NLEFT) and its anthropic implications for the
nucleosynthesis of $^{12}$C and $^{16}$O in red giant stars. We also review the
extension of NLEFT to the regime of medium-mass nuclei, with emphasis on the
determination of the ground-state energies of the alpha nuclei $^{16}$O,
$^{20}$Ne, $^{24}$Mg and $^{28}$Si by means of Euclidean time projection.
Finally, we review recent NLEFT results for the spectrum, electromagnetic
properties, and alpha-cluster structure of $^{16}$O.
