  We single out and study a natural class of Banach spaces -- almost square
Banach spaces. In an almost square space we can find, given a finite set
$x_1,x_2,\ldots,x_N$ in the unit sphere, a unit vector $y$ such that
$\|x_i-y\|$ is almost one. These spaces have duals that are octahedral and
finite convex combinations of slices of the unit ball of an almost square space
have diameter 2. We provide several examples and characterizations of almost
square spaces. We prove that non-reflexive spaces which are M-ideals in their
biduals are almost square.
  We show that every separable space containing a copy of $c_0$ can be renormed
to be almost square. A local and a weak version of almost square spaces are
also studied.
