  We consider group orders and right-orders which are discrete, meaning there
is a least element which is greater than the identity. We note that free groups
cannot be given discrete orders, although they do have right-orders which are
discrete. More generally, we give necessary and sufficient conditions that a
given orderable group can be endowed with a discrete order. In particular,
every orderable group G embeds in a discretely orderable group. We also
consider conditions on right-orderable groups to be discretely right-orderable.
Finally, we discuss a number of illustrative examples involving discrete
orderability, including the Artin braid groups and Bergman's
non-locally-indicable right orderable groups.
