There are 27 Trophies that can be collected.

Bronze

Enter a new challenger…
Play a Quiz, alone or with friends.

Quiz Hero
Score over 1,000 points in a Short Quiz, alone or with friends.

One Small Step
Take your first steps in Explorer.

Child's Play
Complete 5 Quizes on Easy Difficulty (profile owner only).

Mommas and Poppas
Complete 5 Quizes on Hard Difficulty. (profile owner only).

Jiggery-Pokery
Complete your first Jigsaw puzzle.

Squared Away
Complete your first Square puzzle.

Slipping and Sliding
Complete your first Slider puzzle.

Silver

Quiz Lord
Score over 2,000 points in a Medium Quiz, alone or with friends.

Bragging Rights
Win a Long Quiz against friends.

Dr. Livingstone, I presume?
Control all 6 African Territories in a game of Explorer.

Have a Captain Cook!
Control all 6 Australian-Asian Territories in a game of Explorer.

The Grand Tour
Control all 6 European Territories in a game of Explorer.

Silks and Spices
Control all 6 Asian Territories in a game of Explorer.

Voyage of Discovery
Control all 6 North American Territories in a game of Explorer.

The Inca Trail
Control all 6 South American Territories in a game of Explorer.

Prince of Puzzles
Win a Puzzle Battle against another player.

Lord of the Dance
Complete all Jigsaw puzzles. (profile owner only).

Square Eyed
Complete all Square puzzles. (profile owner only).

Let It Slide
Complete all Slider puzzles. (profile owner only).

Counter-Intelligence
You have finished a round in any Quiz with a negative score, alone or with friends.

Gold

Quiz Titan
Score over 4,000 points in a Long Quiz, alone or with friends.

Hearts and Minds
Control half the Territories in a game of Explorer.

On a Roll
Win 5 consecutive, competitive puzzles in a game of Explorer.

All for One…
Complete a game of Explorer with 3 other players beside you.

…One for All
Complete 5 Quizes alone (profile owner only).

Diversity
You captured a Territory in each colour in a game of Explorer.

Platinum

Columbus Salutes You!
Unlock All Trophies in National Geographic Challenge!

http://www.ps3trophies.com/forums/trophy-news/60538-national-geographic-challenge-trophy-list.html