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Pros
Pro Extendable with plugins
Because Backbone is so simple, and the different components are very well isolated, it is very easy to extend the functionality of Backbone. If you're writing your own extended code it's easy to keep it separated out, and share with the community.
The community also has many plugins already available, which you can pick and choose to use to fit your programming style.
Pro Doesn’t force you into a particular coding style or paradigm
There is no “magic” happening below the surface: the source code is clear, readable and well commented. Backbone is also “lightweight” in the sense that it doesn’t require a ton of buy-in to use. It can be easily integrated into an existing page, and you can choose to only use certain components of the library (Views without Models or Collections, for example). While there are many frameworks that seem to be faster to get started with, Backbone’s lack of surprises, clear documentation, speed & flexibility make it a good fit for all types of apps.
Pro Easy to interface with the API
Backbone provides Model and Collection classes that provide strong analogs to restful resources. These strong analogs allow you to interface more naturally with the API, and makes it easier to write custom behaviors for more complicated API interactions.
Collections provide a variety of powerful manipulation methods that are integrated from the underscore library, that allow you to manipulate, sort, and filter collection data easily.
Pro Can be combined with any library you want
Backbone can be combined with any library of your choosing, giving you the ultimate flexibility in creating customized solutions for unique projects. Even the only two backbone requirements, jQuery and underscore can be replaced with equivalent libraries like zepto or lo-dash.
Pro Gives you the freedom to implement views however you want
A lightweight view class is provided but there is no default templating method implemented. Because views are minimal it allows for much more freedom to implement views however you would like, and because of this freedom it's possible to write views to more uniquely adapt to a problem.
User interactions are done within an events object that allows these interactions to be segregated from the rest of the view code which makes the behavioral aspect of the view easier to read and manage.
Cons
Con Requires more coding compared to other frameworks
Because many features are not provided out of the box, you either have to write more base class code to get those features, or find a plugin that provides them in a way you like.
It does not provide much structure either, things like memory management must be kept in mind by the developer, also the lack of view lifecycle management makes state changes prone to memory leaks.