  We prove a Szemer\'edi-Trotter type theorem and a sum-product estimate in the
setting of finite quasifields. These estimates generalize results of the fourth
author, of Garaev, and of Vu. We generalize results of Gyarmati and S\'ark\"ozy
on the solvability of the equations $a + b = cd$ and $ab + 1 = cd$ over a
finite field. Other analogous results that are known to hold in finite fields
are generalized to finite quasifields.
