#Longest production run without gaps

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.

Oldest design still produced, possible gaps in production

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!). Like most general aviation types built in the US, production seems to have paused sometime in the late 1980s to early 1990s, although this is actually pretty hard to verify.

If production didn't pause, the Bonanza definitely takes the prize.