You can download the last version of the app on the website.

If you want to make sure it will always be up to date without downloading it again from the website, you can use the repository.
Download the key :

wget -O - https://dl.itch.ovh/archive.key | sudo apt-key add -

then

# For Ubuntu 18.04
ITCHIO_DEB="deb https://dl.bintray.com/itchio/deb xenial main"
echo $ITCHIO_DEB | sudo tee /etc/apt/sources.list.d/itchio.list

# For Ubuntu 17.10
ITCHIO_DEB="deb https://dl.bintray.com/itchio/deb artful main"
echo $ITCHIO_DEB | sudo tee /etc/apt/sources.list.d/itchio.list

# For Ubuntu 16.04
ITCHIO_DEB="deb https://dl.bintray.com/itchio/deb xenial main"
echo $ITCHIO_DEB | sudo tee /etc/apt/sources.list.d/itchio.list

# For Debian 8.x
ITCHIO_DEB="deb https://dl.bintray.com/itchio/deb jessie main"
echo $ITCHIO_DEB | sudo tee /etc/apt/sources.list.d/itchio.list

# For Debian 7.x
ITCHIO_DEB="deb https://dl.bintray.com/itchio/deb wheezy main"
echo $ITCHIO_DEB | sudo tee /etc/apt/sources.list.d/itchio.list

Once this is done, you can install the app :

sudo apt update && sudo apt install -y itch

Since Ubuntu 18.04, you should use the installer from the official repository and not from steampowered.com.

sudo apt install -y steam-installer

If your steam app is broken after a system upgrade, you have to remove steam and install it again :

sudo apt purge steam* && sudo apt install -y steam-installer

You will have to log yourself in again and if you added some additional game folders, do it again in View → Settings→ Downloads → Steam Library Folder.

Don’t worry, you won’t lose your games 😉