  We show that, in good residual characteristic, most supercuspidal
representations of a tamely ramified reductive p-adic group G arise from pairs
(S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a
character of S satisfying a simple root-theoretic property. We then give a new
expression for the roots of unity that appear in the Adler-DeBacker-Spice
character formula for these supercuspidal representations and use it to show
that this formula bears a striking resemblance to the character formula for
discrete series representations of real reductive groups. Led by this, we
explicitly construct the local Langlands correspondence for these supercuspidal
representations and prove stability and endoscopic transfer in the case of
toral representations. In large residual characteristic this gives a
construction of the local Langlands correspondence for almost all supercuspidal
representations of reductive p-adic groups.
