We show that the number of unit-area triangles determined by a set S of n
points in the plane is O(n20/9), improving the earlier bound O(n9/4)
of Apfelbaum and Sharir [Discrete Comput. Geom., 2010]. We also consider two
special cases of this problem: (i) We show, using a somewhat subtle
construction, that if S consists of points on three lines, the number of
unit-area triangles that S spans can be Ω(n2), for any triple of
lines (it is always O(n2) in this case). (ii) We show that if S is a {\em
convex grid} of the form A×B, where A, B are {\em convex} sets of
n1/2 real numbers each (i.e., the sequences of differences of consecutive
elements of A and of B are both strictly increasing), then S determines
O(n31/14) unit-area triangles