I've seen similar questions to this around various forums, but not precisely what I'm looking for. So, being specific:

When trying to get a CGI script to run from an Apache server, I keep getting a 403 Permission Denied error. All ordinary HTML pages serve up just fine.

The script in question has permission set to 755, which seems to be right. The ScriptAlias is pointing at the right place. Previously, when it wasn't, the page would get served up as text, without executing, but now I have fixed the ScriptAlias issue, I am being denied permission.

Any ideas?

I take it that the script is in the ScriptAlias directory and that you have got all the cgi AddHandler and LoadModule directives in place?

Take a look in the apche error log - it will usually goive you more of an idea what is wrong:
less /etc/httpd/logs/error_log