When we talk about doing mathematics, what comes to our mind is Blackboard-chalk, notebook-pen and books. No doubt that these are and will remain one one of the most important instruments leading to elegant mathematical discoveries. But, the evolution of technology we use has also affected the way we do, learn and share mathematics. In ancient time, mathematics was shared in form of books and letters. Then in 17th Century people started publishing academic journals periodically, which has today become one the most profitable business (like pharmaceuticals). In 1960s computer algebra systems were invented (called MATHLAB). Then in 1970s books were digitized and today we have dedicated ebook readers. Another major challenge of publishing mathematical knowledge was to be able to typeset weird symbols, and this problem was fully solved using computers in 1978 by Donald Knuth. (to know more about this transition read this discussion).
Now it’s 21st century and the shape of sharing mathematical knowledge has changed significantly in past decade. To begin with, in 2003 Poincaré conjecture’s solution was not published in any journal but was rather posted on arXiv. Today we have people on social networking sites like Facebook, Twitter, Google+, Tumblr, Weblogs... who let you know the results just as they are being cooked up. For example, Live-tweet of Babai’s first Graph Isomorphism talk, in this talk one of the most interesting theorem of 2015 was proved. Many big shots announce their big results directly on their Weblogs, for example Terence Tao announced his proof of Erdős Discrepancy Problem on his blog. Today we can have interactive textbooks (like this), articles (like this) and assignments (like this) with advent of MathJax, SageMath…
So far I have been concerned about “print” mathematics, but with advent of cheap internet, whole new methods of mathematical ideas sharing have come into picture. Today almost every reputed research organization maintain video lecture archives (IAS, CIRM, IHÉS, IHP, Institut Fourier, MatScience). Apart from mainstream mathematics, popularization of mathematics has become much more interesting today. We have lots of interesting mathematics popularization channels on YouTube like ViHart, Numberphile, Mathologer, 3Blue1Brown, The Global Math Project,… and SoundCloud like BBC Radio 4: More or Less, ACMEScience ,… For a big-list of online mathematics videos see this and for big-list of mathematics podcasts see this.
Before this internet era, there were similar mathematics popularization attempts. Like my favourite: “Donald Duck in Mathmagic Land” (1959) [updated the link on 28 Dec’18]
But I’m not aware of existence of mathematical radio programs back then. So, if you know about such radio programs,o please let me know about them as comments below.