This paper considers the foundational question of the existence of a fundamental
(resp. essential) matrix given $m$ point correspondences in two views. We present a
complete answer for the existence of fundamental matrices for any value of $m$.
Using examples we disprove the widely held beliefs that fundamental matrices always
exist whenever $m \leq 7$. At the same time, we prove that they exist
unconditionally when $m \leq 5$. Under a mild genericity condition, we show that an
essential matrix always exists when $m \leq 4$. We also characterize the six and
seven point configurations in two views for which all matrices satisfying the
epipolar constraint have rank at most one.