Totally torn cruciate ligaments do not successfully repair themselves.
Partially torn cruciates may repair themselves successfully, but a surgeons may choose to 'augment' a partially torn ligament with a graft so that the patient does not go on to a total tear and secondary damage to other structures.
Early surgical procedures (eg. MacIntosh) relied on strengthening the supports on the outside of the joint (extra-articular repair). A surgical breakthrough was made in the late 1970s with the introduction of synthetic ligament replacements, which led to the development of surgical techniques and instrumentation for internal reconstruction (intra-articular reconstruction).
Intra-articular reconstruction is now mainstream, with the 'hamstrings autograft' or the 'bone-patellar tendon-bone' autograft being the standard procedures. An autograft uses material (usually tendon) from another part of the patient's body for the graft material.
The surgical procedure for a torn ACL is usually a reconstruction using graft material.
Some surgeons do attempt to repair an ACL that has just ruptured (i.e. 'primary repair' - sewing the ends together), but this is not common practice. Usually a ligament replacement procedure ('reconstruction' or 'graft') is performed (i.e. cutting out the old ligament and putting in a replacement).
There are currently a number of types of procedure, each with different good and bad points and different indications, but in practice most knee surgeons will concentrate on getting good at one technique and stick to it.
It is possible to use strong synthetic material to replace the ligament, and also possible to use animal tendon or cadaver (human donor = 'allograft'). But in general tendon material is taken from the patient's own body ('autograft'), usually from the same knee as the injury and the body does a pretty good job of repairing the donor site.
The first choice is usually one or the other of -
BPTB - bone-patellar tendon-bone autograft
(also called PTG - patellar tendon graft)
Hamstrings - four-strand semitendinosus/semimembranosus autograft
In general, most patients in competent hands get good results. But there can be complications (see the section on complications).