  We give general lower bounds on the maximal determinant of n by n
{+1,-1}-matrices, both with and without the assumption of the Hadamard
conjecture. Our bounds improve on earlier results of de Launey and Levin (2010)
and, for certain congruence classes of n mod 4, those of Koukouvinos, Mitrouli
and Seberry (2000). In an Appendix we give a new proof, using Jacobi's
determinant identity, of a result of Sz\"oll\H{o}si (2010) on minors of
Hadamard matrices.
