Compatibility of Comet with current technology
I hear that I can use Comet as a server push technology along with my Ajax code to increase the performance of my web applications.
How mature this Comet technology?
Is it supported by all web servers, programming languages and browsers?
What are the disadvantages of using Comet?
It is mature, though I think you should consider it more of a technique than a technology.
All web servers support it as far as I know, though you will need to research and configure your particular web server if you are building a comet application as the demands on the resources are a bit different. Specifically, there will be far more simultaneous open connections to your server. In terms of programming language support, if your server language of choice has any sort of blocking or waiting mechanism, you can support server-push. All browsers support it as well, as from the perspective of a browser, this is simply an http(s) connection that takes a long time to return.
There are a couple of disadvantages, in the browser world, the biggest is probably the fact that some browsers limit the number of open connections to a specific URL to two. So if you have a server blocking connection open waiting for some pushed data, you are down to only one connection available for the browser to get data from the server. This can be mitigated by spreading your resources over a few second level domains to allow the browser to open more connections.
"Supported by all web servers" is a bit of an odd statement. Most implementations are a server in and of themselves, and you'll need to find a server that integrates with the language you want to use.
That said, I work at a company that built one to integrate with a server, specifically IIS.
If you don't want to bother dealing with the server integration (dealing with different languages, handling scaling, etc), check out websync - the service lets you integrate any language easily, since it's hosted, but supports proxying requests through your own server so you can add your own business logic, logging, permissioning, etc.
Comet was actually in use before all the hype about AJAX started: It's just a new name for an old idea. People have been using hidden iframes to emulate server pushing for a long time without problems.