Cross-platform deployment is surely one of the best features of Libgdx. However, supporting a platform is hard, even when Libgdx does the dirty work through Google Web Toolkit (GWT), which translates your core Java code into obfuscated, optimized JavaScript code. This can be interpreted by any WebGL browsers without the need for extra plugins.
This magical translation comes with a price. GWT does not support every single feature from the Java specification. In order to know whether a feature can be emulated or not, take a look at the official documentation at http://www.gwtproject.org/doc/latest/RefJreEmulation.html.
Another challenge starts once your code is able to be deployed into WebGL browsers—adapt it to the GWT build process.