Here, I start with a key paper on searching for broadcasting civilizations written in 1959, which can be regarded as the theoretical beginning of the age of SETI. We move on from there to the first practical SETI endeavour – Project Ozma. I distinguish between incoming and outgoing signals; the latter are sometimes characterized as ‘active SETI’. Regarding incoming signals, nothing has yet been interpreted as having come from alien life. The LGM signal (little green men) of the 1960s was a pulsar; the ‘Wow’ signal of the 1970s remains enigmatic, but no repeats of it have been observed; the FRBs (fast radio bursts, discovered in 2007) are almost certainly not from biological sources. Regarding outgoing signals, we look at the pioneering Morse and Arecibo messages and the various later ones up to the time of writing (2019). So far, no outgoing signal has been replied to, but this is not surprising since most have not reached their targets yet. Next, I turn to the Breakthough Initiatives, launched in 2015. These include an ambitious plan – Breakthrough Starshot – to send micro-spacecraft to the Alpha Centauri system. I end by briefly looking at ethical issues involved in SETI.