From an earlier answer of mine

One of the classic questions for MA/PhD exams for the aspiring students of modern France is "too what extent was Napoleon the heir of the French Revolution." Although the idea that Napoleon betrayed and ended the Revolution was a common one even among his contemporaries, things were somewhat more complicated in reality. Napoleon went to great lengths to portray himself as an heir of the Revolution and his reign as a necessary reaction to the Terror's excesses. This imperial line was not always very credible and there were other cultural luminaries like Beethoven who were at odds with Napoleon's autocratic turn. Napoleon's proclamation of himself as Emperor did alienate him from Jacobins and other social levelers and ideologues, but it was not as terrible a retrograde step from the Revolution as it might appear at first.

The attitude of various Revolutionaries towards executive power was always very unclear, but there was throughout the Revolution voices that called for strong central order. The experience of the Terror made many within the Directory leery of pure democracy. Pierre Jean Georges Cabanis, one of the leading post-Terror politicians and no friend of Napoleon, said democracy was "odious to all sensible and right-thinking men" and "kept all citizens in a constant state of agitation and anxiety." There was a widespread sentiment among a number of French elites to seek an ordered solution to political power after the Terror. Although subsequent Napoleonic propaganda would emphasize him as the natural and inevitable choice for the Brumaire coup, there were several other French generals various factions were courting for a coup. Emmanel Sieyès, one of the key plotters in Brumaire, would famously quip, "I need a sword," in other words, a man who could command the loyalty of one of the few powerful institutions that possessed a great deal of popular legitimacy within the Republic, the army. Bernadotte, a general and Minister of War, was the favorite of the Neo-Jacobin factions that wanted to reestablish the Revolution's leveling tendencies. Jean-Baptiste Jourdan toyed around with many different factions, but never made a firm commitment. Lazare Hoche, who enjoyed almost as much fame as Napoleon had unfortunately died of tuberculosis before Sieyès's initial soundings towards him could bear fruit. Jean Moreau, the youngest general in the army, refused to compromise himself with politics despite entreaties to do so.

Napoleon's own elevation to Emperor carried with it several important facets that made quite different than his fellow imperial monarchs like Alexander I or Francis I. First off, Napoleon's title, Emperor of the French, carried with it specific, secular connotations and that the title's legitimacy stemmed from the national principle. Napoleon's imperial title was also validated by a constitution and approved of by the populace in a national plebiscite. Although the constitution was rather weak (Napoleon always preferred constitutions to be "vague and short") and the plebiscite rigged, the fact that either occurred showed that this was not simply a return to the pre-1789 status quo. Second, although the Napoleonic regime borrowed many of the trappings of the ancien regime's monarchical style, Napoleon used tradition to sanctify the new regime, not symbolize the return of the old one. Napoleon consciously chose the pageantry of monarchy as a stratagem to curry popular support and feed the notion that the Empire enjoyed true popular sovereignty. Although Napoleon's plebiscites and Senate decrees supporting the Empire were often manufactured, they do show how deeply concerned Napoleon was over the perception he enjoyed popular support. The Revolution's notion of popular sovereignty trumping the interests of the monarch came to be Napoleon's undoing in 1814. After observing the abortive Malet conspiracy of the previous year, Talleyrand noticed that the Imperial Senate did have the power to legally depose of Napoleon. Although the Senate had often been a servile rubber stamp for Napoleon's autocracy, it had the legal potential to declare a new state. Talleyrand, still a Senator, convened the legislative body after he had sounded out like-minded conspirators in his salon, and declared a provisional government.

Although it is tempting to write off Napoleon as a self-serving reactionary, Napoleon really did implement and modify elements of Revolutionary government and political thought into the fabric of the First Empire. The ideal of an Enlightenment-inspired order and a rational and secular state administration had great currency during the Revolution and found fuller expression within the Empire. Even the Concordat with the Papacy fit within this search for order; French priests were to be servants of the secular state which in turn created a protracted struggle between church and the French state that continued throughout the nineteenth century. Napoleon's erection of client kingdoms instead of republics was cut from much of the same cloth as his republican predecessors. Napoleon certainly added a more personalist tone to this process as he put members of his own family on the various thrones of Europe. Yet even this was more of an evolutionary outgrowth than a departure from prior practices. The "sister republics" of the Republican France were frequently inconstant allies of French and they often did not put up much of a fight when France's enemies occupied them. Imposing direct French rule but mediated through local organs would rectify this situation, at least in Napoleon's formulations. The satellite kingdoms continued Revolutionary practices of streamlining administration and setting it up on a more rational basis, albeit often with a member of the Bonaparte clan as its new sovereign. Napoleon's instructions to his relatives often urged them to rationalize their domains and impose a degree of order over them. Although the Napoleonic regime jettisoned much of the rhetoric surrounding republican freedom, the content of French policy remained much the same. The view from Paris, whether Directory, Consulate, or Empire, for these states was consistent: French interests were inseparable from European ones and these states owed France for their modernization.

Although European left-wing politics in the nineteenth century would make much of Napoleon's alleged ending or betrayal of the Revolution after Brumaire, this breakage with the Revolutionary past was more apparent than real. The French Revolution was a multifaceted political animal and Napoleon was able to tap into certain elements of French and Enlightenment politics to justify his dictatorship. Yet, as his ignominious disposal in 1814 by the Senate and his appeals to the Revolutionary liberalism in the Hundred Days show, Napoleon could not ignore the various political ideals and concepts the Revolution unleashed in 1789 despite his pretenses of being a figure above politics.