After Napoleon was FINALLY defeated at Waterloo in 1815 he abdicated for the second and last time. European nations then convened the Congress of Vienna and reestablished the monarchy in France with Louis XVI’s brother who became Louis XVIII. (Louis the XVII was left open as the infant son of Louis XVI who died in prison). So the Cliff notes version is: In 1798 France started as a monarchy which was followed by the Revolution and the First Republic which was followed by the Empire under Napoleon until his first defeat and then a monarchy again. Then Napoleon returned from his exile at Elba for 100 days which ended at Waterloo and then the monarchy ruled again starting in 1815. In 1848 the French revolted yet again and established  the Second Republic. They elected as their president the nephew of Napoleon (!) who decided a few years later to overthrow the Republic and reestablish the empire with him as Napoleon III. This lasted 18 years before he was finally deposed and the Third Republic was formed and this one lasted until the invasion by Germany in the Second World War.

(The Great Courses – Living the French Revolution and the Age of Napoleon)