Here are some critical facts about Luxembourg:

- Luxembourg is a small landlocked country bordered by Belgium, France, and Germany in western Europe.

- Luxembourg has a population of over 600,000 in an area of about 2,586 square kilometers. Its capital and largest city is Luxembourg City.

- Luxembourg has three official languages: French, German, and Luxembourgish. Luxembourgish is based on the Moselle Franconian dialect group.

- Luxembourg has a highly developed economy focused on banking, steel production, high-value-added business services, and international trade facilitated by its large port and airport.

- Politically, Luxembourg is a parliamentary democracy and constitutional monarchy, established initially as a grand duchy in 1815.

- Over 90% of Luxembourg's population is ethnically Luxembourgish, French, German, or Belgian. Historically, in a Catholic country, about 70% of citizens practice Catholicism today.

- Luxembourg enjoys one of the world's highest GDP per capita. It also has one of the smallest carbon footprints per capita worldwide.

- river valleys carve Luxembourg's rugged, hilly countryside. Its climate is moderated, transitional between oceanic and continental.

So, in summary, Luxembourg is an affluent Francophone country that focuses on globalized business and trade and has a high standard of living and income.