Do you really need a VPN?

Over the years, many security experts, and more than a few non-experts, have said that a VPN is an essential piece of security software for travellers and the users of remotely-accessed business systems. When I entered the corporate world in the 1990s, remotely accessing office systems without a VPN was considered a massive risk. But is that still the case? What are the cases for and against using a VPN and is there still a place for them?