Catherine tells Orlov that she has met René Descartes. This is one of the "occasional" parts of the story that isn't true; Descartes was long dead before Catherine was born.
Peter is depicted as already emperor when he marries Catherine. In reality, they married while Peter's Aunt Elizabeth was still Empress and were married for over 15 years before Peter became Emperor.
Catherine recommends that the court ladies read Rousseau's book "The Social Contract". However, Catherine married Peter in 1745 and "The Social Contract" was not published until 1762.
At the party towards the end of the episode (where the Bear is killed), just before Peter (Nicholas Hoult ) and friends are talking about the sites on the guns, a crew member can be seen in the middle of the scene, (clearly identifiable by his modern jacket and ear piece).