The current version of ASDF is ASDF 3.1. It actually contains two systems: the runtime support layer UIOP and the defsystem itself ASDF/DEFSYSTEM.
On all actively developed implementations, you can (require "asdf") and the implementation will provide ASDF 3.1. ASDF is thus provided by ABCL, Allegro, CCL, CMUCL, ECL, LispWorks, MKCL, SBCL. ASDF also works with the now inactive or mostly dead implementations CLISP, CormanLisp, GCL, Genera, MOCL, RMCL, SCL, XCL; their latest release may contain an ancient ASDF (or not ASDF at all), but you can install a recent ASDF on top.
ASDF will not automatically download software. For that, use Quicklisp. Do NOT use asdf-install anymore, for it is both obsolete and unmaintained. There is an unrelated project, asdf-vm, that might attract clicks away from this project.