The Beggar's Opera, as a satire, spends a large part of its energy attacking the broad social injustices of poverty and corruption in eighteenth century England. In addition to the direct attack on corruption and poverty in the abstract, Gay sets up strong references to several individuals who he saw as influential in creating and maintaining these evils, or representative of their workings and effects. The most direct and most striking of these references are those to Robert Walpole, a prominent English statesman, and Jonathan Wild, the notorious thief-taker and criminal hanged in 1725.
Robert Walpole was one of the most powerful men in Eighteenth century England. Though he left few changes on the body of English law behind him, his political power and skillfull maneuvering within the English parliament causes him to be credited as the man who invented the position of Prime Minister. A more complete overview of his political career may be found here, but in brief, Walpole held a variety of offices, rising steadily from treasurer of the Navy through the house of commons, until becoming first lord of the Treasury and chancellor of the Exchequer in April 1721. He held these positions and maintained his political supremacy until 1742. Despite his political prowess, he had many enemies. In addition to the political battles between the Whigs and their Tory opposition, he made several moves later in his career that alienated the London merchants and the house of Lords, chief amongst them a proposed excise tax of 1733. Most germanely for our purposes, Walpole alienated the literary intelligensia, among them the Scriblerians, an association of classically educated writers that included such figures as Jonathan Swift, Alexander Pope, and John Gay. Bolingbroke's Opposition paper, The Craftsman, was among the most vehement and consistant sources of satyric attacks on Walpole, and likewise associated with the literary circle formed around Pope and Swift. In fact, although it was directed primarily at Henry Fielding, the licensing act passed in 1737 can also be read as a sign of Walpole (and his establishment) trying to control through force of law the literati, made newly powerful by the advent of mass printing, and lacking the social constructions that limit the virulence of such.
Jonathan Wild moved in a very different circle than Walpole, but was equally well-known in London, or perhaps moreso. At the time, the policing of London was carried out by thief-takers, who usually started out as thieves themselves, and who were paid £40 for each conviction that their arrests led to. Wild, who declared himself "Thief-Taker General," took advantage of this system to an astonishing degree, operating as both sides of the law. He ran a gang of criminals who would both commit crimes themselves (primarily theft) and apprehend the members of other criminal gangs for Wild to send to the gallows. The loot from these heists would go into one of several warehouses that he owned. At the same time, he ran a business where he claimed to find and return stolen goods, allegedly procured in the process of thief-taking. Of course, they really came straight from the warehouse where his gang dropped them off. The genius of the plan, and the way in which Wild kept control of his gang, is that, if any of the gang members seemed likely to rebel against his power, he could simply have them hanged; for who had more evidence to arrange against them than the man who gave them their marching orders. In this fashion, he held an almost complete monopoly on the London underworld, while remaining an influential man in conventional London. It was not until he was caught returning stolen property to someone directly (while in Newgate on another charge) that he was finally hanged in 1725. His skeleton is still on display at the Royal College of Surgeons' Museum. The Wikipedia entry for him has useful information.
Despite the vast difference in the means in which they came to power, Walpole and Wild had in many respects similar methods of operation. Both were expert at using various legal tricks and loopholes to gain and hold power; both were famously corrupt sexually as well as politically (Macheath's relationship with both Lucy and Polly is thought to be a dig at his mistress, Molly Skerrit, who later became his wife). Most importantly, both were positioned at the head of vastly powerful hierarchical structures, which influenced the lives of every Londoner, from beggar to king. The Beggar's Opera, produced a year after the death of George I (a crisis moment for Walpole), and three after the execution of Jonathan Wild, was excellently timed to draw the public's eye. Additionally, these men were linked in a discourse of the time, that of the "Great Man." A key element of the bourgoise/noble frictions, the contention that one could be great, that is, powerful and influential, without also being good, was thrown into sharp relief by both the ascendance of Walpole and the actions of Jonathan Wild. This anxiety forms the central part of several texts, and forms a key element of The Beggar's Opera. Throughout the play, the word 'great' is used as a synonym for 'powerful' or 'effective,' and deliberately linked with success or power in criminal or corrupt endeavors.