Twiliobot is a robot built using the Twilio API that extends Google Wave with telephony capabilities. When added to a wave, twiliobot automatically transforms phone numbers into click-to-call links, allowing users to make calls and connect with each other in real-time. The subsequent conversation can be recorded, transcribed, and inserted into the wave as text with a link to the audio of the conversation. This integration enables powerful voice applications using Twilio with Google App Engine.