This is a very common question so I wanted to answer it here.
You can stream with any device as long as your video is encoded in H.264 (audio in AAC) and it supports RTMP publishing to Wowza/Adobe FMS (generic streaming services). With the exception of USB webcams (which have other downsides), using your camera for streaming can present a few challenges.
Unfortunately, a GroPro does not work as a USB webcam. For a device it function as a USB webcam it must be detected as a video capture device when connected via USB, otherwise known as USB streaming. As user friendly as it may be in other areas, you can not just plug it into your machine and start streaming with it.
HDMI streaming (applies to other cameras as well):
You can use an HDMI capturing device to capture the HDMI output but your live encoding software will need to have support for this device. This is the most straightforward way to use a GoPro to stream but it does require a device like the BlackMagic Intensity. There are other devices but if it's not specifically supported by your live encoding software then it won't work. Wirecast has the best range of support for these devices but X-Split also has support for some. Be aware of the Windows version of the Intensity, it requires a specific USB 3.0 chipset.
Another option is to use a dedicated streaming device instead of a computer and HDMI capturing device combined. This makes things a lot easier but it's not cheap.
Gotcha: You'll also have to make sure USB is not connected to your computer at the same time since it will disable HDMI out, you'll need a standalone power supply or run off the battery.
Wireless streaming (works but far from ideal):
The GoPro also has wireless streaming so you can access the stream from a PC, however using this to publish a stream is very kludgy.
You'll have to be on the same network as the GoPro, determine it's IP address and then access the m3u8 playlist that represents the stream. You'll then have to open that stream up in something like VLC and play the video back on your PC. Finally you'll use your live encoding software to capture the part of your screen that is playing the video back and use that as the video source, so basically it's doing a live video screen capture of just the part of the screen showing the VLC player.
This will of course create issues if you are trying to capture audio, so you'll need special software to route the audio of the VLC player to a virtual audio device that you can then use as an input in your live encoding software. Virtual Audio Cable for Windows or Sound Flower for OS X.
Here is a tutorial on how to get the wireless streaming working. The instructions are for Twitch but the same applies to our service as well. Note that these instructions lack the extra steps necessary to get audio working properly using Virtual Audio Cable or Sound Flower.