At what age can children start doing simple puzzles?

0 years. The whole world is a puzzle to them. The only thing that's age-dependent is size of puzzle pieces -- they should not be such that the child can choke on them, below 3 years of age, or breakable.
1-2 years. Most babies can start with simple shape sorters before their first birthday. Simple puzzles with basic shapes can be done as early as 1-2 years, and watching your child advance to more complicated puzzles can be very rewarding. Some children master this early on, others take longer. If you have any concerns about her development, speak to your pediatrician.