Vladimir S. posted the correct solution, although he originally didn't link directly to the section that had the answer. The direct link is here.
For those who'd rather not click, here's the official solution per Microsoft:
Debian and Ubuntu based distributions
The easiest way to install for Debian/Ubuntu based distributions is to download and install the .deb package (64-bit) either through the graphical software center if it's available or through the command line with:
sudo dpkg -i <file>.deb
sudo apt-get install -f # Install dependencies
Installing the .deb package will automatically install the apt repository and signing key to enable auto-updating using the regular system update mechanism (apt update). Note that 32-bit and .tar.gz binaries are also available on the download page.
The repository and key can also be installed manually with the following script:
curl https://packages.microsoft.com/keys/microsoft.asc | gpg --dearmor > microsoft.gpg
sudo mv microsoft.gpg /etc/apt/trusted.gpg.d/microsoft.gpg
sudo sh -c 'echo "deb [arch=amd64] http://packages.microsoft.com/repos/vscode stable main" > /etc/apt/sources.list.d/vscode.list'
Then update the package cache and install the package using:
sudo apt-get update
sudo apt-get install code # or code-insiders