How can I declare and use Boolean variables in a shell script?

Revised Answer (Feb 12, 2014)

the_world_is_flat=true
# ...do something interesting...
if [ "$the_world_is_flat" = true ] ; then
    echo 'Be careful not to fall off!'
fi

Original Answer

Caveats: https://stackoverflow.com/a/21210966/89391

the_world_is_flat=true
# ...do something interesting...
if $the_world_is_flat ; then
    echo 'Be careful not to fall off!'
fi

From: Using boolean variables in Bash

The reason the original answer is included here is because the comments before the revision on Feb 12, 2014 pertain only to the original answer, and many of the comments are wrong when associated with the revised answer. For example, Dennis Williamson’s comment about bash builtin true on Jun 2, 2010 only applies to the original answer, not the revised.