There is now finally an official upgrade mechanism for Homebrew Cask (see Issue 3396 for the implementation)! To use it, simply run this command:

brew upgrade --cask

However this will not update casks that do not have versioning information (version :latest) or applications that have a built-in upgrade mechanism (auto_updates true). To reinstall these casks (and consequently upgrade them if upgrades are available), run the upgrade command with the --greedy flag like this:

brew upgrade --cask --greedy

To get outdated:

brew outdated --cask --greedy --verbose

As of December, 2017, brew cask upgrade is a thing.

https://github.com/Homebrew/brew/pull/3396

As said by Melonee in the comments, Homebrew Cask by default won't automatically update apps with built-in update function. If you want to do so you have to run:

brew upgrade --greedy

According to man brew:

--greedy
       Also include casks with auto_updates true or version :latest.

Beware that in some rare case, applications will store user data in the .app folder. Upgrading them via Homebrew Cask will uninstall and reinstall them, deleting associated user data. (See comments)

Casks that have auto_updates true are not included in brew outdated or brew upgrade by default as upgrades are handled by the applications themselves. If you would like them to brew included you can pass the --greedy-auto-updates flag to the commands.

If you're only trying to upgrade one package:

brew upgrade package_name

All packages

brew update && brew outdated && brew upgrade && brew cleanup

Note, I had to add my own answer for this since I got here looking for just one, and noticed the top answer had brew upgrade --cask, while I got the following from here:

During 2020 the sub-commands of the cask command were gradually integrated into the core of Homebrew and were deprecated. The core commands now operate on both formulae and casks. To limit their scope to only formulae or casks add the --formula or --cask option to the command line.

The cask command has been completely removed from Homebrew in version 2.6.0, released in December 2020.

There is a unified flow now for both formulae and casks:

brew update
brew outdated
brew upgrade

The first command (brew update) gets from the Homebrew servers the information about the most recent versions of the available formulae and casks.

The second command (brew outdated) lists the outdated installed formulae and casks. Add --formulae or --casks to the command line to limit the command scope to formulae or casks. Add --greedy to the command line to include in its output the casks that are marked as "auto-updateable". This command is not needed. It can be used to see what programs will be upgraded before doing the actual upgrade.

The third command (brew upgrade) upgrades the packages listed by brew outdated. It also supports the options listed above for brew outdated and operates accordingly.