Twitter has made it easier to add rich, responsive Tweet displays directly to your website or CMS. There are now 3 options:
1. Factory Functions - Allows you to easily generate timelines for any web app.
2. oEmbed API - Integrate profile, list, like, or collection timelines directly into your CMS.
3. publish.twitter.com - A website that lets you customize the feed and get the code you need to copy and paste into your site.
List template example:
Grid template example:
Timeline types - User timeline:
Find more examples of what you can do at publish.twitter.com here.
Per the Twitter blog:
As part of these improvements, we’ve also removed the need to create and save widgets to your account. Of course, if you already have timeline widgets, you can always access them in your Twitter settings. But, going forward, you won’t need to log in to create or configure embedded timelines, and you’ll never need another widget ID again. All you need is a public profile, list, like, or collection URL to get started.