  In a celebrated paper of 1893, Hadamard established the maximal determinant
theorem, which establishes an upper bound on the determinant of a matrix with
complex entries of norm at most $1$. His paper concludes with the suggestion
that mathematicians study the maximum value of the determinant of an $n \times
n$ matrix with entries in $\{ \pm 1\}$. This is the Hadamard maximal
determinant problem.
  This survey provides complete proofs of the major results obtained thus far.
We focus equally on upper bounds for the determinant (achieved largely via the
study of the Gram matrices), and constructive lower bounds (achieved largely
via quadratic residues in finite fields and concepts from design theory). To
provide an impression of the historical development of the subject, we have
attempted to modernise many of the original proofs, while maintaining the
underlying ideas. Thus some of the proofs have the flavour of determinant
theory, and some appear in print in English for the first time.
  We survey constructions of matrices in order $n \equiv 3 \mod 4$, giving
asymptotic analysis which has not previously appeared in the literature. We
prove that there exists an infinite family of matrices achieving at least
$0.48$ of the maximal determinant bound. Previously the best known constant for
a result of this type was $0.34$.
