  We show that the number of unit-area triangles determined by a set $S$ of $n$
points in the plane is $O(n^{20/9})$, improving the earlier bound $O(n^{9/4})$
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 $\Omega(n^2)$, for any triple of
lines (it is always $O(n^2)$ in this case). (ii) We show that if $S$ is a {\em
convex grid} of the form $A\times B$, where $A$, $B$ are {\em convex} sets of
$n^{1/2}$ real numbers each (i.e., the sequences of differences of consecutive
elements of $A$ and of $B$ are both strictly increasing), then $S$ determines
$O(n^{31/14})$ unit-area triangles.
