The Wire app appears, on the surface, to avoid many of the pitfalls that other messaging apps have - it's fully end-to-end encrypted, and it's open source. It supports video calling, and it has clients for smartphones, desktops, and web. Unlike e.g. WhatsApp, you can use the web client even when your smartphone is off.
On their privacy page, they state that
Text messages and pictures use Off-the-Record (OTR) end-to-end encryption. Wire uses the Axolotl ratchet and pre-keys which are optimized for mobile messaging.
However, Moxie Marlinspike has said that they do not use the Signal protocol:
Wire does not use Signal Protocol, they used some of our code to create a protocol of their own devising that we do not recommend.
I'm unable to find any other details on this. Is there any commentary on Wire's encryption? Is Wire secure?
Edit 2017-05-17: it looks like Wire stores a list of everyone you've communicated with in plaintext on their server, but this may change in the future.