I love asyncio! It's a very well put together library. It provides great interfaces to manage event loops, io, and some basic networking. It gives you a lot of freedom to design asynchronous systems as you see fit.
However, batteries are not included. For example, it provides no HTTP client/server. It doesn't interop with any synchronous IO tools in the standard library either, making asyncio a very insular environment.
For the majority of problems, Go or Node.js may be better options. They have much more mature environments for managing asynchrony.