The title of Monaco's ruler Prince Albert II is Prince of Monaco, and the conundrum starts there. The prince also styles himself Duke of Valentinois, just one of many French titles the Grimaldi family lays claim to. There lies the second conundrum. The Prince and other family members are addressed as His (or Her) Serene Highness, and that’s a third conundrum.
Rainier I Grimaldi started out as a Genoese pirate in exile taking the castle of Monaco in military action. He ousted its Genoese garrison and owners in 1297. He was soon chucked out again in turn. In 1331, his son Carlo Grimaldi retook the castle; he officially received the castle from the French King Philip VI the Fortunate together with Menton and Roquebrune and was styled Seigneur de Monaco. Over the next 500 years, the Grimaldi family of Monaco collected numerous French and Italian titles and lands from various thankful monarchs.
In 1457, Claudine of Monaco set the precedent for female succession by assuming the lands and titles in her own right and styling herself as Dame de Monaco. In 1612, Seigneur Honoré II elevated himself to Prince of Monaco with the backing of the Spanish King Philip III and nobody really cared to contest the issue. As a prince, he would be addressed in French as Son Altesse Sérénissime which translates to His Most Serene Highness but this translation is customarily reserved for ruling princes of the Holy Roman Empire. For that reason, family members today have to be content with HSH (His or Her Serene Highness); it’s an unjust world.
Due to the precedent set by Claudine, the French lands and titles remained in the family for centuries despite a second female succession happening with Louise in 1731. Claudine had married a Grimaldi cousin, but Louise was married to Jacques de Goyon-Matignon, Count of Thorigny and Matignon, Duke of Estouteville who took on the name Grimaldi and the family titles and reigned in lieu of his wife. He set the precedent for a groom changing the name if marrying the prospective heiress of Monaco.
During World War I, France became progressively worried about the situation in Monaco. The ageing Prince Albert I had only one son called Louis. And that son didn't show any intention to marrying and begetting children. If they both died, the German Duke of Urach (a cadet line of the Royal House of Wurttemberg) would inherit the principality. France could have German submarines landing in a port on its Mediterranean coast any day after that event.
Prince Louis had an illegitimate daughter named Charlotte Louvet who was unable to succeed him for that reason. The French government had a clear idea of what they wanted, and it was something Monaco could live with considering they were defended by the French army. France and Monaco made a treaty whereby France guaranteed Monaco’s independence for as long as a Grimaldi from the direct line of the ruling Prince would sit on the throne. If the line failed, France would inherit Monaco to become part of France.
Prince Louis then went on and adopted his illegitimate daughter Charlotte Louvet officially in Paris with the signature of the French Prime Minister sealing the precedent that adopted succession is accepted as direct line succession by France. The precedent holds true for applied Monegasque succession laws, too. Charlotte assumed the title of Princess of Monaco by adoption and Duchess of Valentinois by investiture by Prince Albert. The French titles, by the way, are contested by French purists who hold that titles may not pass through illegitimate offspring or adoption. They may be right. On the other hand, wily Prince Albert had included her titles in the adoption documents signed by the French President which basically can be construed that France accepted them as being held under Monegasque law.
Somewhere along the line, the French had lost their Royal heads. And therefore nobody could tell Charlotte that she was not Duchess of Valentinois. Viewed from today, the titles therefore passed through usus into Monegasque law, and the Prince may do with them what he wants; they are definitely hereditary under Monegasque law. Should France ever return to a monarchy, the future king or queen may well bestow the title of Duc du Valentinois where they please; we might come to the point where titles have to carry an add-on of (French) or (Monacan). The Problem seems not imminent, though.
The titles used for Prince Albert II are: Duke of Valentinois, Marquis of Baux, Count of Carlades, Count of Polignac, Baron of Calvinet, Baron of Buis, Lord of Saint-Remy, Sire of Matignon, Count of Thorigny, Baron of Saint-Lo, Baron of La Luthumiere, Baron of Hambye, Duke of Estouteville, Duke of Mazarin, Duke of Mayenne, Prince of Chateau-Porcien, Count of Ferrette, Count of Belfort, Count of Thann, Count of Rosemont, Baron of Altkirch, Lord of Isenheim, Marquis of Chilly, Count of Longjumeau, Baron of Massy, Marquis of Guiscard.
The treaty with France was broadly worded; not to be restricted too much to the persons currently involved, it names only the direct line of the Prince of Monaco as eligible. The first to notice that this formula was a mistake was Rainier III. Upon assuming the throne, he found that the formula had overnight excluded his sister and her children from the succession. It was obvious that that had not been the intention of the signatories. They had wanted to make sure the Duke of Urach could not inherit; they had not intended to put pressure upon every future prince to instantly produce or adopt an heir. The amendment of 1961 of the treaty was therefore aimed at naming all children of Rainier III as part of the succession, thereby taking off the pressure from Albert II to marry and have children; he made ample use of that.
The new treaty of 2001 which scrapped the inheritance clause completely (Monaco would even remain independent as a Republic). The Principality was able to rephrase the inheritance laws to ‘normal’ dynastic succession thereby restoring Princess Antoinette and her children to the succession. In its wake, Princess Caroline’s title should be correctly given as Hereditary Princess of Monaco as she is currently the successor to Prince Albert II. Nobody bothers with it, though.
Princess Caroline decided that her children from her marriage to Stefano Casiraghi would be better off without titles. They are therefore plain Andrea, Pierre, and Charlotte Casiraghi. No princes or princesses there. Her fourth child Alexandra in turn is a princess; by dint of being the daughter of Ernst-August Prince of Hanover she is Alexandra Princess of Hanover. She is no Princess of Monaco, though. You might have noticed the difference in the location of titles. Princess Caroline as opposed to Ernst-August Prince of Hanover. The clue lies in history. Caroline holds the personal title of Princess of Monaco and the territorial title of Hereditary Princess of Monaco. Ernst-August on the other hand has the family name of Prince of Hanover and no title. Titles went out of fashion in Germany in 1918 when they were abolished and converted into family names.