  We derive a formula for the number of pre-images under a non-degenerate
harmonic mapping $f$, using the argument principle. This formula reveals a
connection between the pre-images and the caustics. Our results allow to deduce
the number of pre-images under $f$ geometrically for every non-caustic point.
We approximately locate the pre-images of points near the caustics. Moreover,
we apply our results to prove that for every $k = n, n+1, \ldots, n^2$ there
exists a harmonic polynomial of degree $n$ with $k$ zeros.
