JQuery is an easy to use library for Javascript. It is not a framework, nor is it scalable to write an entire application within it.
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Pros
Pro Easy to use
Much easier to use than traditional JavaScript. Allows a lot of features and function to be completed with less lines of code.
Pro Cross-browser support
jQuery works on all modern browsers, although there may be some issues with old versions of each browser.
Pro Strong community
The open source community has really embraced jQuery and it has many readily available plugins to help speed up the development process. In addition to this, it has great documentation and tutorials
Cons
Con It is not a framework, but rather a library.
Frameworks allow you to handle page routing, authentication, and services. jQuery is not a framework, but rather a library of DOM tools for websites.
e.g.: You cannot write page routing logic within jQuery.
Con Script download
As opposed to just using raw JavaScript, users will (automatically) download the jQuery script to their local computers in order for it to work. This results in more overhead on the initial page visit, regardless of how lightweight the framework is.