Longest production run (civil)

The Beechcraft Bonanza seems like a likely candidate for 'oldest design still being made'; first flight was 1945, and it's still in production today (albeit made by the same parent company that makes Cessnas!). Even during the general aviation downturn in the US of the 1980s & 90s, production continued, as Kevin Cathcart pointed out in the comments:

FAA registrations show no gap in years of manufacture of the A36 Bonaza. I was able to find a registration for one manufactured in every year from 1980-1999 in california alone. Production did drastically slow during that time frame, but it never stopped.

#Longest production run (military)

The Lockheed C-130 Hercules military transport first flew in 1954 and has been in continuous production ever since. According to Wikipedia, they've even been built in the same factory (in Marietta, Georgia) the entire time.

Because only around 2,300 have been produced (only!), they don't show up on the 'most produced aircraft list' mentioned in another answer.