Very popular in French & other European cultures, the doudou is a transitional object, it, in short, is a substitute for the mother when she is not around.
According to studies, we start seeing the doudou appear between 4 to 6 months. It is at the age of 8 or 9 months that the child starts carrying and asking for his doudou, an age at which he has difficulty 
being separated from his parents.
The doudou is ever present at night and when he finds himself alone at nap time.
THE ROLE OF A DOUDOU:
When the child realizes that he and mommy are 2 separate people, fear of losing her arises and the doudou helps to comfort and reassure. Its familiar scent helps the child cope with separation anxiety from the parents.
CARING FOR THE DOUDOU:
After several month of constant use, some parents may be embarrassed by the state the doudou is in. Not to worry, a real doudou often looks worn because your child carries it everywhere he goes.
To avoid altering the familiar smell, you must never throw away or change a doudou without your child’s permission. The mother should hug the freshly washed doudou for a few minutes to re-establish her scent.
Having back-up doudous in case one is lost is also prudent. In some rare cases children may refuse a substitute, which is painful for parents and child. But by alternating them on a regular basis, the risk of rejection is greatly reduced.
THE SEPARATION:
Parting ways with a doudou is not an easy task. Your child will progressively detach himself from his doudou all by himself. He will become more interested in socializing with other children. Between the ages of 3 and 5 years, the doudou may only be a sleepytime friend.
Source: La science des doudous par Marine Cygler, Le monde de l’enfance Oct-Nov 2007