  Signatures are primarily used as a mark of authenticity, to demonstrate that
the sender of a message is who they claim to be. In the current digital age,
signatures underpin trust in the vast majority of information that we exchange,
particularly on public networks such as the internet. However, schemes for
signing digital information which are based on assumptions of computational
complexity are facing challenges from advances in mathematics, the capability
of computers, and the advent of the quantum era. Here we present a review of
digital signature schemes, looking at their origins and where they are under
threat. Next, we introduce post-quantum digital schemes, which are being
developed with the specific intent of mitigating against threats from quantum
algorithms whilst still relying on digital processes and infrastructure.
Finally, we review schemes for signing information carried on quantum channels,
which promise provable security metrics. Signatures were invented as a
practical means of authenticating communications and it is important that the
practicality of novel signature schemes is considered carefully, which is kept
as a common theme of interest throughout this review.
