For Mac users - a new webcam app.

I'm adding this to existing 'webcam' questions for future searchers.
I am in no way affiliated with this product or the company making it - this is a simple user to user recommendation.

I just discovered this today, announced on DPReview a new product called Cascable Pro Webcam £40 [£30 for the first week to 24 July 2020]

I've just tried the demo & it "just worked".
I've never had anything before on Mac that could do this with my D5500, so I thought I should let others know about it.
The app runs you through a setup procedure, including asking if you want to connect over WiFi or USB. I went with USB for my Nikon D5500.
Switch camera off, connect to USB, switch on - camera is discovered & set up automatically. It then asks if it can install a plugin to add it as an input option on any camera-capable app.
Done, that's it.

Here's their list of compatible cameras

If you have a Canon or Nikon camera, and a Windows PC, Sparkocam might be a solution. It enables you to connect the camera via USB, and use the camera for for example Skype.

I have used my Canon EOS 700D with SparcoCam, and it works well. It is free to try, but costs $49.95 for a standard license for either Canon or Nikon (upgrades for 1 year).

From Sparsoft.com:

How to Use DSLR Camera as a Webcam

If you don't have a laptop with a built-in webcam or an external webcam or simply want to make the most of your digital camera capabilities (like aperture adjusting for blurring your background), there is a way to adapt your digital camera as a webcam.

Some digital cameras have ability to be used as webcams and are shipped with necessary webcam software, but most camera vendors don't provide such software. In this tutorial you will learn how to use Canon DSLR or Nikon DSLR camera as a regular webcam with the help of SparkoCam. SparkoCam is the software that allows you to turn your Canon or Nikon camera into a standard webcam for video chats, conferencing, recordings, etc.

inkista has an excellent answer above.

My Panasonic GX-80 doesn't support 'Lumix tether for streaming' - but it does have a Wifi function allowing you to watch a live preview from a smartphone.

Over the course of a caffeine-fueled long weekend, and helped by the writings of people who'd tried similar things before me, I was able to get the wifi preview stream functioning as a webcam under Linux.

The result is only a 640x480 video stream, it's not very reliable if there's wifi interference, and it's delayed by ~100ms. But compared to the laptop webcams used by 99% of people it's still a pretty big step up in quality.

Obviously there are only so many Panasonic users on Linux, so it's unlikely my tool can help you personally - but if you're a computer programmer, and your device has a preview mobile app, you too may be able to bodge it into service as a webcam, by reverse-engineering the preview video stream.

There are three main methods by which a dSLR/mirrorless camera can be used as a webcam.

HDMI capture

HDMI capture requires two things: that the camera can output "clean HDMI"; that is HDMI that doesn't have camera control overlays all over it, and an HDMI capture device to turn the HDMI output from the camera into a USB webcam input to the computer. Most* video-capable interchangeable lens cameras do not output clean HDMI if they are not from the last generation or two (this is written in 2020) of cameras.

But HDMI capture does allow for HD resolutions and 4K.

*_{Canon dSLRs that can load Magic Lantern excepted. See: https://magiclantern.fandom.com/wiki/HDMI_Output}.

Tethered liveview translation

The second method is using a camera's ability to do tethered liveview over USB, and using software to translate this to look like a USB webcam input. Resolution may be limited.

Also, a lot of camera bodies can't do this. Canon and Sony excepted, most camera manufacturers reserve this feature for higher-end prosumer models. So you'll need to check whether your specific camera has this capability. You then need some software that can translate this liveview stream to look internally like webcam input. Typically, transforming it into Spout (Windows), Syphon (OSX), or V4L2 (Linux) input, and then software to make that a virtual camera you can select from your video conferencing app.

Many camera companies are rolling their own versions of how to do this (see By camera manufacturer section, below), but there are free and open solutions that can work for older cameras and OS versions (see my personal experience section).

Screen capture streaming

The third method is to use some kind of USB (or wireless, see mjt's answer) connection with the camera and software that can display the camera sensor's data (typically a remote control shooting app from the manufacturer, or freeware like digicamControl or gPhoto) and then using OBS (Open Broadcast Software) to send that display on a webcam stream using obs-virtual-cam or its plugin (Windows), obs-mac-virtualcam (OSX), or obs-v4l2sink (Linux).

This method typically still requires USB liveview capability, but doesn't rely on having a newer camera or OS versions that are supported by the OEM webcam releases.

By camera manufacturer

Nikon

Nikon has released the Webcam Utility for Windows 10 and OSX 10.13, 10.14, and 10.15.
It lists the following supported cameras:

• Z 7II, Z 7, Z 6II, Z 6, Z 5, Z 50
• D6, D5
• D850, D810, D780, D750, D500
• D7500, D7200,
• D5600, D5500, D5300, and D3500

Sony

Sony has released their Imaging Edge Webcam software for Windows 10. It supports 35 models. Petapixel notes:

Supported cameras include every Sony a7 and a7R since the Mark II, all three a7S cameras, both a9 cameras, and the a5100, a6100, a6300, a6400, a6500, and a6600.... The software also supports three A-mount models, and a slew of RX-series cameras as well.

And they give a complete list here. The A6000 is not on it.

Enthusiasts have also published how to do screencap streaming and HDMI translation with Sony cameras. The chances of getting clean HDMI or liveview tethering with Sony is pretty high.

Canon

Canon has released the Canon EOS Webcam Utility for Windows 10, OSX 10.13, 10.14, and 10.15 for the following cameras to take advantage of liveview tethering:

• 1 DX II, 1DX III
• 5D Mark IV, 5DS, 5DS R
• 6D Mark II
• 7D Mark II
• 77D, 80D, 90D
• SL2, SL3
• T6, T6i, T7, T7i, T100
• M6 II, M50, M200
• R, RP,
• G5X II, G7X III, SX70 HS

This reddit post lists it as also unofficially working with:

• T5i, T4i, T3i; and (on this SE, the T2i)
• T6s
• 60D (no C-AF), 70D
• 5D Mk II (no C-AF)

So, if your Canon dSLR isn't listed, try it anyway and see. You may need to be in video mode.

Fuji

Fuji has released X Webcam for Windows 10 and OSX 10.12-10.15, and lists compatibility with the following cameras:

• GFX100, GFX50S, GFX50R
• X-H1
• X-Pro2, X-Pro3
• X-T2, X-T3, X-T4

And there's a firmware update for the X-T200 and X-A7, so they can be used as webcams without additional software on Windows 10 and OSX 10.14/10.15.

Panasonic

Panasonic has released Lumix Webcam Software (Beta), for Windows 10 and Mac, and lists compatibility with:

• DC-GH5
• DC-G9
• DC-GH5S
• DC-S1, DC-S1R, DC-S1H
• DC-S5
• DC-G100/G110 (in the future)

Olympus

Olympus has released Olympus OM-D Webcam Beta for Windows 10. And lists compatibility with:

• E-M1X
• E-M1, E-M1 Mark II, E-M1 Mark III
• E-M5 Mark II

They also have a page on using an HDMI capture device, and lists the same models as having clean HDMI output, excluding the E-M1 (Mark I).

Android/iOS smartphone cameras

Apparently, smartphone cameras can also be used as webcams, if you had need for more video-studio style manipulation vs. simply using the conferencing app for your phone. They typically do something similar to the liveview tethering deal, only using a mobile app, wi-fi, and a Windows/OS driver or application to turn the wifi signal into a webcam. See:

• Wired article: "How to Turn Your Smartphone into a Webcam"
• Tom's Guide "How to use your phone as a webcam".

My personal experience with an older camera and OSX 10.12 (Sierra)

However, if you're an OSX shooter and your camera and OSX version are so old they aren't on the lists above or can use obs-mac-virtualcam, I've also found Kit Farrelly's free solution for OSX tethered liveview translation, which uses the open source Camera Live to turn the tethered liveview into a Syphon stream, and CamTwist to turn the Syphon stream into a webcam feed. I wrote about that in this answer on a Canon-specific question. Farrelly got it to work with his Fuji X-T2; I use it with a Canon 5DMkII. The resolution with my 5DMkII (confirmed with Syphon Simple) client is 1024x680.

I've also found that the $20 commercial eCamm application, iGlasses, can turn a Syphon feed into a webcam feed and supports basic adjustment controls/effects, if you didn't want to purchase an eCamm Live subscription, deal with Camtwist [abandoned after 2013], or you needed a solution that works with FaceTime [Camtwist doesn't].

See also:

• How can I set up a Canon EOS camera to be a webcam?, particularly Linux users: see confetti's gphoto2/v4l2loopback-dkms answer), which is not Canon-specific.
• How to use your DSLR Camera as a Webcam In Linux, another liveview solution which uses gphoto, v4l2loopback, and ffmpeg.
• eCamm Live, SparkoCam, and vMix commercial packages
• Use any camera as a webcam. A Youtube video on screen capture streaming. However, "any camera" is misleading, since you have to have some way to get liveview from your camera to display on your computer.