In the past I have use ASIHTTPRequest but now there is NSURLRequest. Should we be using the NSURLRequest now? Are there any disadvantages
NSURLConnection and NSURLRequest are the provided Cocoa classes for managing connections. In iOS 7, Apple added NSURLSession.
But I think you'll find AFNetworking to be a framework that further simplifies network requests (notably complex HTTP requests). If you don't want to use third-party frameworks for any reason, you can use NSURLConnection and/or NSURLSession directly. It just takes a little more coding.
If you're looking for a good ASIHTTPRequest alternative, AFNetworking is a great candidate.
For information on NSURLConnection and NSURLSession see the URL Loading System Programming Guide.