Downtown is a good place to hunt for souvenirs. The World Trade Center Souk is a modern version of a traditional market while Madinat Zayed Shopping & Gold Centre collects the city's gold jewellers all under one roof. Elsewhere, you'll find standalone small boutiques and art galleries speckled through the neighbourhood.
Practical Tip: Shopping for Gold
Although the style of gold jewellery displayed in the gold souq of Madinat Zayed Shopping & Gold Centre isn't to everyone's taste (think heavily ornamental yellow gold), for those who are eager to shop rather than simply browse, handcrafted pieces can be bought for a fraction of what it would cost back home.
This is because gold (here usually of around 23 karat) is bought by weight while the cost of the fine handicraft of each piece of jewellery is usually free. So when you buy bangles, necklaces and other items by weight, no extra cost is added for the complexity of design or quality of workmanship. Check the price of gold online before you go to get a rough idea of the expected price.
Note that in Islamic tradition, it is haram (forbidden) for men to wear gold – they usually wear silver.