The people of Luxemburg speak Lëtzebuergesch (Luxembourgish), which was officially designated as the national language in 1984. This is a Germanic language descended from a Moselle-Franconian dialect, but having numerous French loan words, particularly in the fields of business and administration. Many German speakers can understand spoken Luxembourgish to a greater or lesser extent.
Until recent years Luxembourgish was predominantly a spoken language with no standard orthography. However, there is now a vigorous campaign to promote its use and a rapidly increasing number of books and newspaper articles are published in Luxembourgish. Luxembourgish is also spoken to some extent in the adjacent Belgian province of Luxembourg, despite official discouragement there.
The official languages are French and German. Inhabitants of country areas tend to be more proficient in German. Town dwellers are usually more or less at home in both languages, though often with a slight preference for French. Laws are published in French, although parliamentary debates are conducted largely in Luxembourgish, with some speeches in French (particularly formal statements by members of the government) and German. The most widely read newspaper (the Luxemburger Wort) is mainly in German, although it also carries some articles in French and a few in Luxembourgish. Other influential newspapers include the Tageblatt (published in German) and the Luxembourg edition of the French Républicain Lorrain.
Although Luxembourgish is used increasingly for teaching, particularly for younger children, the principal medium of instruction in primary schools is German, as children learn to read and write first in German. German continues to be used to a lesser extent in secondary schools, where French is the predominant language.
I wrote the above text for my 1994 book European Financial Reporting: Luxembourg. One important change since then is the founding of Luxembourg University in 2003. The book sold 400 copies, doubtless not separate sales but sales of the whole series, sponsored by the Institute of Chartered Accountants in England and Wales.
Luxembourg University provides resources on Lëtzebuergesch at Luxembourgish (uni.lu)