This library is by far the most popular component library out there (if Github stars are any indication of the popularity of a repository). I tested it out when it was still newly developed and found it promising. Since that time it's had quite a few releases and has become a full fledged library with lots of features. This library is great if you want your application to have Google's Material design or if you don't have a designer and want your application to have a polished look and feel.
React Toolbox is another UI component library that implements Google's Material design. The cool thing about this library is that it uses CSS modules. With material-ui, to provide any customization you either need to override the style prop or use a global object to override styles for your whole project. With react-toolbox, every component has a "theme" prop that takes a CSS Module object. If you want to override styles for a particular component, you import your custom .scss file. Changing global styles (colors, fonts, animations, etc) means overriding the global.scss file variables and styles with your own custom theme.scss file.
Elemental is another UI library of components that is great for when you need some basic building blocks without a lot of styling. Elemental doesn't make too many assumptions about the look and feel of your application. If you have a custom application with a lot of distinctive designs, then this would be a great library to use. Think of it as Bootstrap or Foundation, but built from the ground up with React.
This library, while a little smaller than the other libraries we've featured (in terms of components), has some interesting components that aren't available in the other libraries. It includes components such as a toggle button that works on mobile and a star rating component
Grommet is another full featured UI library that also has some interesting components not found in the other UI libraries. Aside from the typical buttons, typography, and other common UI elements, it has a World Map component, a video player, a carousel and simple charts. A lot of the components would be great in a dashboard or used in an admin section of a site.
Oy is a library that allows you to render email templates on the server. The blog post announcing Oy describes some of the motivations behind the creation of it. For example, React only allows HTML5 tags and attributes, and for backwards compatability reasons, some HTML4 tags and attributes are necessary in email clients. The Oy library allows for these attributes.
Once you've got your application looking great, how do you ensure future developers and designers keep things consistent and up to your standards? While there is no silver bullet to ensure everything in your application looks pixel perfect and consistent, having a style guide can help. With this library you can create documentation for every type of component in your project.
React-Flexbox-Grid is a set of React components that implement flexboxgrid.css, which is a CSS grid library based on the flex display property. It's built on top of CSS Modules, Webpack and ES6.