The Counts of Luxembourg were a reknown family of the medieval nobility. Emperor Charles IV. of the Holy Roman Empire was one of the Counts of Luxembourg and in 1354 he elevated the county to a dukedom for his brother Wenzel, who had inherited Luxembourg. Wenzel was childless, so Charles IV. set him up to adopt his son and heir Wenzel as heir to Luxembourg. But nephew Wenzel was childless, too, and had struggled for much of his life with his brother Sigismund for the crown of the Holy Roman Empire. So on his death, he willed Luxembourg to his younger brother John and his heirs. Not that it helped John, as Sigismund controlled Luxembourg.
In 1437 the last Count/Duke of Luxembourg (and king of Bohemia and Hungary), Emperor Sigismund of the Holy Roman Empire died without a male heir and left his lands to his daughter princess Elisabeth of Bohemia and Hungary, the wife of Albrecht, duke of Austria. Thus Luxembourg became part of the Habsburg-lands with Albrecht becoming Roman-German king in 1438.
But that was quite short-lived. Sigismund's brother John, duke of Görlitz had left the claim to Luxembourg to his only child, duchess Elisabeth of Görlitz and Luxembourg. The duchess married twice, first to Anthony of Burgundy, duke of Brabant and second to John of Bavaria, count of Zeeland and Hainault who owned a large part of the Netherlands.
As she was childless herself, she contracted an inheritance treaty with her nephew by marriage Philip the Good duke of Burgundy, who took Luxembourg in 1443 after paying the duchess a large sum for her claim, so the duchy of Luxembourg became part of Burgundy. Two generations later the heiress Maria of Burgundy brought Luxembourg and the other parts of Burgundy (including the Netherlands) to her husband, later Holy Roman Emperor Maximilian of Habsburg in 1477.
In 1555 Luxembourg went to the Spanish line of the Habsburgs, was involved in several inheritance wars and became part of the Austrian Habsburg-lands in 1713. Napoleon annected Luxemburg for France, but in 1815, at the Congress of Vienna, Luxembourg was elevated to a Grand-Duchy and given together with the Crown of the newly formed Kingdom of the Netherlands to the former Prince of Orange, who had fought successfully against Napoleon. The difference was that while the Netherlands still retained their former structure of the Nassau-Orange as "firsts among equals", Luxembourg was considered the private property of the king of the Netherlands. But different to the Netherlands, Luxembourg had been part of several treaties including rules of inheritance between the branches of the House of Nassau - when Wilhelm III. (of Nassau-Orange) died in 1890, leaving only a daughter, Wilhelmina became queen of the Netherlands but Luxembourg was inherited by her male cousin Adolf from the Nassau-Weilburg-line. That happened even though Luxembourg as well accepts female inheritance.
Hope this helps.
If you are interested who still carries the blood of the Counts of Luxembourg, here's a list: Stammliste des kaiserlichen Hauses Luxemburg â€“ Wikipedia