In the late 18th century — as America was on the cusp of revolting to free itself from the tyranny of arbitrary rule by people born to privilege and laws, medicine and freedoms constrained by religious superstition — Europe was, state by state, wrestling with many of the same issues. It was the Enlightenment, the Age of Reason, and if it’s a sketchy historical period for most Americans, the Danish drama “A Royal Affair” makes a stately, entertaining way of bringing us up to speed.

It’s a tragic romance, a tale of idealism, usurped power and reform ahead of its time, of palace intrigues, madness and forbidden love. And aside from the odd detail here and there, it’s mostly true.

In the 1760s, young Princess Caroline (Alicia Vikander) has been groomed in Britain to marry handsome King Christian VII of Denmark. She’s heard he loves the arts. She learns she has been misinformed.

Christian (Mikkel Boe Folsgaard) is bratty, temperamental and immature.

He does love the theater, especially Shakespeare, whom he memorizes. He’s a bit mad but still functional enough to serve the purposes of the entrenched nobility that runs the government and his court, including his scheming stepmother, the Queen Dowager (Trine Dyrholm). Caroline, who has a dewy, mournful Emily Blunt quality thanks to Vikander, is trapped in a miserable marriage, surrounded by enemies.

When the king sets off on a long tour of Europe, Caroline is given some peace. And the king, thanks to the machinations of some exiled nobles who want a friendly voice to wrangle them back onto the court, is given a new physician. Johann Friedrich Struensee, played with smoldering earnestness by Mads Mikkelsen, is “an Enlightenment Man,” a healer who treats his work as a higher calling.

Struensee indulges the monarch’s penchant for prostitutes, drink and brawling. He becomes confidant and confessor, and turns Christian into a more reasonable, manageable mess. He could even be a help to the morose Caroline, something he suggests to Christian. “Make her fun,” the king commands. “I want a fun queen.”

But as the queen and her doctor bond over his seditious enlightenment book collection, Struensee provides a kind of fun the king did not have in mind.

Co-writer/director Nikolaj Arcel, who scripted “The Girl With the Dragon Tattoo,” tells this story in a languorous fashion, pausing to let the doctor and the queen find their chemistry on the dance floor, then in the queen’s bed chambers. He takes the time to show the slow, idealistic machinations of the queen and her doctor, who took more and more power at court and used it, high-handedly, to bring Denmark into the Enlightenment.