Most revolve around his boyish appearance, diminutive stature, and reputation as a puppet of President Vladimir Putin, who is expected to become prime minister after stepping down in May.
What is "nanotechnology" in Russian politics? goes one joke. Answer: When each new leader is shorter than his predecessor.
In another, Putin takes Medvedev out for dinner. Putin orders a steak. "What about the vegetable?" the waiter asks him. "The vegetable will have a steak, too," answers Putin.
Perhaps the most popular gag so far is a video that lampoons the cult Soviet film "Kavkazskaya Plennitsa," or "Caucasus Prisoner." The clip has been viewed more than 1 million times on the video-sharing website YouTube.
In the original scene, a local tricks a naive Russian tourist into kidnapping a young woman by persuading him that abductions are a local tradition. In the overdubbed spoof version, the tourist is Medvedev and he's being asked to snatch not the woman but the Russian presidency.
"So what's my role?" asks Medvedev. "Collect signatures, put forward your candidacy," the local tells him. "My candidacy?! That's also part of the tradition? Brilliant!" exclaims an enthusiastic Medvedev.
The local then explains that things must look "natural" even if the vote is bogus. "The candidates will resist, kick, even bite, call for observers, shout that they're going to complain to the United Nations," he says. "But don't pay attention -- it's all part of a beautiful old tradition!"
No Laughing Matter
The Kremlin, if aware of the clip, is unlikely to be amused.
Such jokes are a far cry from the reverent portrayal of Medvedev and Putin in Russia's state-controlled media.
Yuly Gusman, humorist and head of the Russian Film Academy, says the Kremlin has become particularly intolerant of political satire under Putin's tenure.
"It's disappearing from the social sphere and from television debates. It's gone, and I think it will be gone for a long time. I think this began under Vladimir Vladimirovich [Putin]," he says. "[Former President Boris] Yeltsin can be reproached for many things, but he attached great value to freedom of speech and of the press, which attacked him, bit him. He ground his teeth but bore it all."
Gusman knows this from firsthand experience.
The television station covering the Nika film awards ceremony -- Russia's equivalent of the Oscars -- in Moscow last month cut out part of his speech, in which he made a tongue-in-cheek remark about the Putin-Medvedev power tandem.
"Traditionally we have a message from the Russian president," he told the star-studded audience. "Since clearly no one knows who our president is, you can consider it coming from me."
A spoof film sequence shown to Nika guests in which Putin and Medvedev were portrayed as a tsar and his son was also skipped over in the television report.
The Kremlin, however, has tolerated -- sometimes even encouraged -- mild political satire.
As Russians cast their ballots on March 2, national television broadcast images of Medvedev attending KVN, a hugely popular student humor show dating back to Soviet times. Medvedev was shown smiling good-naturedly as some of the performers poked gentle fun at his sudden rise to power.
Yuly Gusman (right) says, "Yeltsin can be reproached for many things, but he attached great value to freedom of speech and of the press" (AFP)"It's meant to show that we supposedly have democrats in our country, freedom of speech, not like under Comrade Stalin," says Gusman. "It's approved humor; it's very pleasant, but that's all."
The more scalding satire that flourished in the 1990s following the Soviet collapse has all but vanished. "Kukly," a satirical puppet show, was one of the first television programs axed after Putin became president in 2000.
"Kukly" showed Putin in a variety of guises, from an indecisive leader struggling to choose a new prime minister to an impotent young king on his wedding night.
One of the most caustic episodes was based on "Little Zaches Called Zinnober," a tale by 19th-century German author E.T.A. Hoffmann in which a wicked dwarf bewitches a city, tricking its residents into worshiping him as a great ruler.
The show depicted Putin as the wicked, foul-mouthed dwarf, and oligarch Boris Berezovsky -- the former Kremlin kingmaker who eventually fell out with Putin -- as the magic fairy helping him put the nation under his spell.
"In the outhouse, waste them all! Waste everyone in the outhouse!" screams the Putin puppet at the start of the episode, in which he is still a baby in his crib -- a reference to Putin's famous statement about Chechen separatist rebels.
"Lie still, boy. Now we'll make a human being out of you," says the Berezovsky-fairy.
The episode enraged Putin's supporters, a handful of whom published an open letter calling for the show's authors to be prosecuted.
The man behind "Kukly," the well-known humorist Viktor Shenderovich, was not dragged to court. But he left the show in 2001 after the television station broadcasting it was taken over by a state-controlled company. A significantly tamer version of "Kukly," without the Putin puppet, was finally ditched in 2003.
Since then, says Shenderovich, political satire has been driven underground.
"It's not the 'Kukly' program that disappeared, it's the country in which the 'Kukly' program existed that disappeared," he says. "Beyond the realm of the Internet, we've returned to a Soviet model. Biting personal satire, the kind that goes for the most sensitive spots, is absolutely impossible in the federal framework. It escaped entirely to the Internet; it escaped entirely to folklore."
Russia has a rich tradition of political satire. For centuries, sharp-tongued "chastushki" -- short, rhymed folk songs -- were devised to ridicule the country's rulers.
Russians also used "lubki," hand-colored folk prints, to poke fun at the elite. Peter the Great, whose drive to secularize and Europeanize the country earned him many foes, was a favorite target of lubok artists. Famous lubki portray him as a cat or a crocodile.
While biting satire has long been an inherent part of political life in Western countries, Russia's history of despotic regimes means this type of humor has almost always been confined to the private realm.
Precisely because it is so deeply entrenched in the Russian psyche, the Kremlin is unlikely to ever succeed in uprooting political satire.
"In Russia, mocking authorities is an exclusively oral, folkloric tradition," says Shenderovich. "There were jokes about Stalin under Stalin's rule; there have always been jokes about leaders, there have always been harsh chastushki, there has always been an entire world of satire -- including satire that is foul-mouthed, popular, vulgar, but incredibly talented. This has always existed, and will always exist."
By Claire Bigg, www.rferl.org