Comet Observatory

The Comet Obervatory is the main hub of Super Mario Galaxy. From here you may access the many galaxies in the game using domes. It is also where Rosalina and her Lumas live. It is first accessed through Gateway Galaxy. A new dome is opened every time you collect a Grand Star (with the exception of beating The Center of the Universe level). Once five Grand Stars are collected and you have 60 or more stars in total, you may travel to the Center of the Universe to save Princess Peach.

Domes

There are six domes in the Comet Observatory. They are unlocked in the following order:

• Terrace
• Fountain
• Kitchen
• Bedroom
• Engine Room
• Garden

Unlocked alongside the Garden is Gateway Galaxy, which can be accessed through the house on the small planetoid across from the Garden. Also, each dome has its own Hungry Luma, which will unlock one additional galaxy for each dome if you feed it.

Hidden 1-Up Mushrooms

• In the crate in the garage (ground point on it or spin next to it to break the crate)
• Under the stairs leading to the Kitchen dome
• Below the core, underneath the library between the Kitchen and the bedroom. You can either long jump to it from the green platforms, or use a Red Power Star
• At the top of the Observatory (only accessible with a Red Power Star)
• Underneath the entrance to the Gateway To The Stars (also requires Red Power Star)

Appearances

Outside of Super Mario Galaxy, it has made appearances in 

• Super Mario Galaxy 2, as a part of the ending
• Mario Kart 7, with Rosalina's Ice World