The Sources: Placing Sankara in a period according to the modern calendar is a difficult problem. The official date accepted currently is 788-820 CE, and the Government of India celebrated the 1200th anniversary of Sankara's birth in 1988. This date is largely based upon one traditional view prevalent in India.  However, the date is still open to question, as pointed out by swAmI tapasyAnanda in his translation of the mAdhavIya Sankaravijayam.  This difficulty is experienced for almost all personalities in Indian history, due to paucity of proper records and conflicting traditions current in different parts of the country. As far as the problem of dating Sankara is concerned, our sources of information are: internal evidence from Sankara's works, the astronomical details recorded in some of the Sankaravijayams, and the traditional accounts kept in the advaita maThas in India.
Internal Evidence: Of these three sources, a lot of scholarly work has been done in the recent past, analyzing the internal evidence from Sankara's works. The date now seems to be converging to the early 8th century CE.  The most important internal evidence comes from Sankara's verbatim quotation of dharmakIrti, the buddhist logician. Hsuan Tsang, the Chinese pilgrim, who visited India in the time of harshavardhana, king of Thanesar (606 - 647 CE), gives clues to dharmakIrti's date. He also mentions bhartRhari, but not of Sankara. It follows that Sankara is post-dharmakIrti, and possibly post-Hsuan-Tsang also. Critical academic scholars are converging to a date near 700 CE for Sankara's period.
Astronomical Details: The astronomical details in the various Sankaravijaya texts are not of much use. More often than not, the details in one work contradict those in another, and one cannot rely on any of them unless one is preferentially biased to accept one of the Sankaravijayas as more authoritative than the others. Dates ranging from the 5th cent BCE to 8th cent CE have been calculated on the basis of such astronomical details. One further complication is that some astronomical information is said to have been obtained from works which are not available anywhere in India. So it is difficult even to authenticate the astronomical details from their supposed sources. Also, not all the currently available texts titled Sankaravijayaare accepted as authoritative within the living advaita tradition. Under the circumstances, it should be noted that the astronomical references in one text is only as good or as bad as all the other such details in other texts, and no firm conclusion can be drawn about their validity.
Records of maThas: Whether Sankara established any maThas at all has been questioned in the modern literature. Thus, Paul Hacker attributes the tradition of four AmnAya maThas at Sringeri, Puri, Dvaraka and Joshimath to vidyAraNyasvAmin. The native oral tradition, however, takes the history of these four maThas, each associated with one of the four geographical directions and one of the four vedas, to SankarAcArya himself. The daSanAmI sannyAsI sampradAya, with its various akhADas in northern India, accepts affiliation only with these four maThas, though such affiliation is largely nominal. There seems to be some historical evidence for the existence of the oldest daSanAmI akhADas as early as the 9th cent. CE.  However, as swAmI tapasyAnanda points out, the evidence of the daSanAmI sannyAsI tradition has never been properly taken into account in the modern literature. It seems very likely that the tradition of four AmnAya maThas reflects historical fact. It is immaterial whether Sankara established them himself or whether these four maThas developed naturally at the places where the four famous disciples of Sankara lived and taught. It is clear that even if they were not actually established by Sankara himself, the four AmnAya maThas came into existence early in the history of post-Sankaran advaita vedAnta.
Of these four maThas, the Joshimath title had long been vacant, till it was revived in 1940 CE. Consequently, it does not have many ancient records. The Dvaraka and Puri maThas have, in the past, claimed a date of 5th century BCE for Sankara. This is partly based upon a dating of a grant by a king named sudhanva who is supposed to have been a contemporary of Sankara. Nothing else is known about this king, and the grant itself has not been dated with any accuracy. In any case, it should be remembered that the records of the Dvaraka and Puri maThas are rather fragmentary, because they have had patchy histories, with periods when there were no presiding SankarAcAryas. This is also accepted by the administrations of these institutions, and they do not hold to the 5th century BCE date with absolute certainty. Meanwhile, Sringeri has been the only maTha of the original four which has had an unbroken succession of maThAdhipatis. This may be no more than an accident of history, as southern India has not experienced as many political upheavals as the north. Given these facts, among the traditional sources, only the Sringeri records seem to lend themselves to critical historical analysis.
The Sringeri maTha's record states that Sankara was born in the 14th year of the reign of vikramAditya. The record does not give any clue about the identity of this king. Some 19th century researchers identified this king with the famous vikramAditya of the gupta dynasty, thereby postulating a date of 44 BCE for Sankara. A period of more than 700 years was then assigned to sureSvara, because the later successors in the Sringeri list can all be dated reasonably accurately from the 8th century downwards. This is rather anomalous, and can be resolved quite neatly, as pointed out by Mr. B. Lewis Rice in his Mysore Gazetteer. 
If one identifies the vikramAditya as a member of the Western cAlukya dynasty, which ruled from bAdAmi in Karnataka, one gets a much more reasonable date for Sankara. The cAlukya dynasty reached its greatest fame in the time of pulakeSin II, a contemporary of Harshavardhana. According to historians, there were two kings named vikramAditya in this cAlukya dynasty - vikramAditya I ruled in the late 7th century CE, while vikramAditya II ruled in the early 8th century.  However, there is still some ambiguity with respect to which of these two vikramAdityas is actually meant, but as with most Indian historical records, this is the best one can do. It is more reasonable to identify the vikramAditya of the Sringeri record with one of these two cAlukyan kings, who ruled from Karnataka, rather than the northern gupta king, whose empire did not include southern India. This interpretation of the Sringeri record is also consistent with the internal evidence from Sankara's works. In either case, this implies that the earliest date that one can postulate for Sankara has to be in the late 7th century CE. swAmI tapasyAnanda also quotes a letter from Sringeri, which makes it clear that this maTha claims nothing more than what its record states, interpretation of dates being the historian's job.  This is the sensible approach to take, given the fact that traditions in India tend to be rather ambiguous in their chronology.
In addition to these four original maThas, a number of other advaita maThas have come into being over the centuries, some of which are quite well-known. These maThas either started out as branches of the original institutions, or were set up as independent monasteries by notable sannyAsIs of the daSanAmI order. With the proliferation of such maThas came a number of "traditions," many of them conflicting with one another in details. For example, some of these maThas also claim to have been established by Sankara himself.  Some of them also claim 5th century BCE to be the date of Sankara.