The Great portrays the marriage between Catherine and Peter as a fraught arranged marriage, and in that regard, it's fairly accurate. Many of the details, though, have been intentionally fictionalised to tell a different kind of story. Most notably: in the show, Catherine arrives, meets Peter for the first time, and marries him after he's already the established emperor.
In real life, the couple — who were second cousins — met as children and did not get along then. They married as teenagers (she was 16; he was 17) while Peter's aunt Elizaveta still ruled, and they spent nearly 17 years together before Peter became emperor. Their marriage was not a faithful one on either side, and both took multiple lovers, a detail that The Great borrows as well. Catherine was thoroughly unimpressed by Peter, historian Virginia Rounding told Time. "She trained herself, learning and beginning to form the idea that she could do better than her husband."