We generally keep these columns free-standing but last week we made an exception. We talked about the comma in sentences like this: "After we had visited the museum, we had a snack." We said that, contrary to accepted wisdom, this comma has nothing to do with the fact that "after we had visited the museum" is a clause.
On the contrary, we said, it's there because this clause is an adverbial, just like "after our visit to the museum", which isn't a clause but also benefits from a comma: "After our visit to the museum, we had a snack." So here's our promised explanation of adverbials.
An adverbial is a phrase or clause that behaves like an adverb. This means you can usually replace them by an adverb; so you could replace both the adverbials in the museum example by the adverb "afterwards", like this:
"Afterwards, we had a snack."
Here too you can have a comma, so it makes good sense to include adverbs themselves under the umbrella of "adverbial": an adverbial is any word, phrase or clause that behaves like an adverb.
And the comma rule? If an adverbial stands at the start of a sentence, put a comma after it. You might want to refine the rule a bit by letting it depend on the length and complexity of the adverbial, but that's something we're happy to leave to your judgement. In general, if the comma helps the reader to follow your meaning, then it's worth preserving.
The point is that it's only adverbials that allow a comma. If the first part of the sentence is its subject, however complex this may be, it shouldn't have a comma. For example: the comma we've put in this sentence, would be quite wrong.
Why are adverbials punctuationally privileged? Because they're not part of the "core" of the sentence. Long-time readers may remember a little allegory about a grammatical squirarchy in which the verb is the squire, and adverbials are the local village people who are brought in to do odd jobs as needed. (You can find this, like all our other columns, at www.phon.ucl.ac.uk homedicktesTES.htm) The sentence's core is the verb plus all the bits that it really needs to make a sentence - the subject, object and so on. You can put a comma after an adverbial because it's not in the core - the verb's grammar doesn't care about it. That's why it's okay in: "Afterwards, we had a snack", but not in: "We, had a snack" or "We had, a snack." And, as we said at the beginning, that's also why it's okay after an adverbial clause: "After we had visited the museum, we had a snack". Without a doubt, adverbials are a useful part of a smart writer's toolkit.