Drain question

I think I've mentioned before that the drain for our driveway and rear walk sometimes has trouble, well, draining. This was a problem yet again last night, and I'm tired of that little stream of water flowing through our basement when the lake forms against the back wall. It's pretty clearly more than just the grate getting clogged that's to blame, but I've exhausted my limited ability to investigate and fix the problem myself.

So: what kind of contractor should I be calling for this? It sounds vaguely plumbing related, but it falls a bit outside of my vague mental model of what plumbers do. Should I be looking instead for the sorts of folks who could repair the cracked sidewalk nearby at the same time? (And, er, who are they, exactly? I've not yet had to have any serious structural work done around the house.) Advice would be appreciated!