When was the last analog computer developed? What about one that is still in use (as in computation or experiment)? Were there major improvement done from the day digital computers became so popular?
Rutherford journal gives a nice chronology of analog computing up to 1970, when they went out of mass use. "The demise of the analogue computer was a gradual process. Apart from reductions in size and improvements in speed, the machines on sale up until 1970 were all essentially electronic differential analysers." Aside from living relics there has been something of a resurgence in the use of analog computing in the last 10 years however.
Artificial neural networks remained a subject of attention even after 1970s. They would be analog computers if implemented directly, rather than digitally simulated, Wikipedia gives some history. In 1990s Siegelmann showed mathematically that theoretically analog neural networks are strictly stronger than the universal Turing machine (although implementation requires non-physical assumptions).
Back in 1971 Chua suggested based on symmetry arguments that in addition to three known passive elements of electric circuits, resistors, inductors and capacitors, there should be a fourth, which he called memristor (memory resistor). Unfortunately, no physical model was known at the time. Only in 2008 Strukov, Snider, Stewart and Williams found a physical realization of memristor, and pointed out that it explains "hysteretic current–voltage behavior observed in many nanoscale electronic devices". In 2009-2015 researchers used memristors to build artificial neural networks for machine learning and pattern recognition. Nanotechnology is generally expected to make wide use of analog computing.
Another motivation for resurgence of analog is the power efficiency wall for digital computers. "Moore’s law — the maxim that processing power will double every 18 months or so — continues, but battery lives just haven’t kept up". This opened a new niche, "low-power, high-performance computing, especially in areas where the answer doesn’t have to be completely perfect — image rendering for example". US Department of Defense funded UPSIDE program led by Hammerstrom, Intel's chipmaker back in 1980s, "to build analog processors that can do probabilistic math without forcing transistors into an absolute one-or-zero state, a technique that burns energy".
In 2012 Rice University designed a "low-power chip that uses probabilistic computing techniques to do energy-efficient, if occasionally inexact, calculations". At Stanford researchers built biological transistors inside cells, and at MIT created "synthetic analog computers that run genetic machinery", they turn on or suppress a gene that manufactures green fluorescent protein, an optically detectable molecule.
The USS Iowa has mechanical analog computers for firing that survived the 1980-refit of the ship.
An analog computer of everyday use in the XXIst century are the Flight computers used on a wide range of aircraft daily.
Electronical analog computers were used to until the 70s because they're very good at analytics's, but I think even electronic analog computers have now lost the race.