  We develop a general theory of "almost Hadamard matrices". These are by
definition the matrices $H\in M_N(\mathbb R)$ having the property that
$U=H/\sqrt{N}$ is orthogonal, and is a local maximum of the 1-norm on O(N). Our
study includes a detailed discussion of the circulant case
($H_{ij}=\gamma_{j-i}$) and of the two-entry case ($H_{ij}\in{x,y}$), with the
construction of several families of examples, and some 1-norm computations.
