The latest instalment, in a case that has lasted for more than four years, comes almost nine months after Judge Thomas Penfield Jackson ruled that the company had violated antitrust laws and ordered it to be split in two.
Microsoft is appealing against the order to divide the company for two reasons - the first being its belief that "neither the facts nor the law" justify the company's break-up; and secondly, it claims that Judge Jackson is biased against the company.
The software giant has argued that comments Jackson made to members of the press in relation to the case show that he was prejudiced.
In one interview, for the New Yorker magazine, Jackson called Microsoft founder and chairman Bill Gates arrogant and compared him to Napoleon.
The court will hear today both from Microsoft and the US Department of Justice on the broad charge that Microsoft had abused its monopoly power. It will seek to establish whether Microsoft illegally linked its Internet Explorer browser with its Windows operating system in an attempt to crush competition from Netscape's Navigator browser.
The second of the two-day timetable of discussions will look at whether a break-up of the company is the appropriate action; as well as try to determine whether Jackson's comments to reporters criticising the company demonstrate bias.
The appeals court is not expected to rule before the summer.
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