How can developers edit a Google Doc programmatically? Is there a Docs API?

The Document List API has been deprecated since September 2012 and looks like it could be retired after April 2015.

Updating the HTML version using the Drive API, as the question suggests, looks to be the only other way. I have been trying this and I have experienced a few of issues.

1. Comments are converted into citations and added to end of document.
2. If someone else is editing the doc via the browser any changes made by them between the API read and update time are lost.
3. Updates to a doc can break the formatting. For example I updated a doc several times and the vertical spacing between some elements (h1's, h2's etc) kept widening each time and ruined the doc.
4. When an API update occurs the cursor of anyone in the doc is moved to the top of the page.

There may be more issues. These are just the ones I have found in the last few days.

UPDATE (May 2019) The Google Docs API was officially launched in Feb 2019. The documentation is located at the link from my update in July below. A few weeks after launch, I produced a high-level video overview of what a mail merge application using the API would look like. (It's not a full-fledged G Suite Dev Show episode but does link to a working sample.)

UPDATE (Jul 2018) The Google Docs team pre-announced a forthcoming REST API at Google Cloud NEXT '18. Developers interested in getting into the early access program for the new API should register at https://developers.google.com/docs. The original answer below still stands as the REST API will become the second way you can access Google Docs programmatically.

Original answer (Mar 2017): (Most other answers are outdated.) Google Docs does not currently have a REST API, however developers can programmatically access (CRUD) documents using Google Apps Script, server-side JavaScript apps that are hosted at and run in Google's cloud. If you're new to Apps Script or to editing Google Docs with it, here are some learning resources:

• Your first script which creates & edits a Doc, then uses Gmail to send it to you.
• I've got 4 intro videos for you (mostly Sheets-flavored)
  • They're in this playlist (see videos 5, 8, 22, 24)
  • Any forthcoming videos will be in this series
• Useful pages in the official docs
  • How to CRUD Google Docs with Apps Script overview page
  • Extend Docs functionality by creating Docs add-ons via this quickstart
  • Apps Script reference documentation for Google Docs (Document Service)
• See Google Docs add-ons that other developers have built

Simple example: if you have an existing Doc with a (Drive) file ID of DOCUMENT_ID_GOES_HERE, here's how you'd basically edit it with Apps Script, doing a pseudo "mail merge" of name & email into the document given placeholders {NAME} and {ADDR}:

function mergeNameEmail() {
    // Open a document by ID
    var doc = DocumentApp.openById(DOCUMENT_ID_GOES_HERE);

    // Access the body of the document
    var body = doc.getBody();

    // Merge name & address from template
    body.replaceText("{NAME}", "Ima Developer");
    body.replaceText("{ADDR}", "123 Main St, Anytown, XX 00000");
}