I've heard about AppleScript Studio many times, but I cannot find any download links.
A tutorial said that I could launch Xcode and select AppleScript under Application when starting a new project, but there wasn't anything listed, which had anything to do with AppleScript.
How can I download AppleScript Studio?
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AppleScript Studio is no longer included in Xcode as of Xcode 4. But there is AppleScriptObjC, a framework which lets you write Cocoa apps using AppleScript.
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Also, if you don't already have Xcode, download it from here: http://developer.apple.com/xcode/. As mentioned above, AppleScript Studio doesn't exist anymore as Xcode supersedes it.
AppleScript Studio was the name used to describe creating an AppleScript application using Xcode, not a separate application. Apple no longer uses the name, and it is called a Cocoa-AppleScript application.