Advantages of using React
React is mostly used on the client-side, but if e.g. SEO is your big concern, you can actually run React on the server. And, as if this was not enough, with little extra effort you can leverage React Native to build native iOS, Android and Windows Phone apps! Also, a viable alternative for developing mobile apps is Apache Cordova and plain React (basically a web-app running inside a WebView).
Our experience with React
There are very few libraries that can really match or challenge React when it comes to ease of composing complex UIs and performance. The learning curve for developers is also pretty tolerable (I'm looking at you, AngularJS).
At rspective, we find React a perfect fit especially for complex UIs - that's where blazing performance really kicks in, and composable architecture pays off in overall simplicity and a solution that is easy to maintain. On the other hand it fits well into small apps too - as they tend to grow with new features over time, so it often makes sense to put React in place at the very beginning, rather than back off from e.g. jQuery or rewrite the whole thing later on.
Another advantage that we see in practise is that it's only a View library. In effect, it doesn't impose how you are supposed to design your entire application, but it gives you freedom to pick your tools of choice and use them together with React as you want. In other words - it doesn't force you to build the app within the constraints of the framework (taking it to the extreme - you don't need to work around your framework), but the framework is there to help you out with the tech so you can truly focus on the business.
Together with libraries that help you handle state through-out your application, like redux or MobX, React positions itself as a state-of-the-art and truly powerful tool. With a great developer community and cross-platform capabilities, it becomes a de-facto standard of the modern web.