A large cohort study from the United Kingdom with 11,101 term infants and 778 preterm infants shows improved cognitive development when infants are breastfed.  J Pediatr 2012; 160: 25-32.  There have been a number of previous studies as well that have shown that breastfed children have an IQ that is ~5 points higher than nonbreastfed children; however, when adjustment for confounders, especially maternal education, this effect is weaker.  Since this is an observation study (randomized blinded study would be impossible), it is difficult to control for all variables.  Nevertheless, prolonged breastfeeding, more than 2 months in preterm and more than 4 months in term infants, is associated with higher cognitive development.

Additional references:

• Am J Clin Nutr 199; 70: 525-35.
• BMJ 2006;  333: 945.
• Arch Gen Psychiatry 2008; 65: 578-84.
• Paediatr Perinat Epidemiol 2003; 17: 81-90.

Other refences:

• J Pediatr 2009; 155: 421.  Breastfeeding may have possible protective effect from developing inflammatory bowel disease.  Review of multiple studies.
• J Pediatr 2002; 141: 764.  Breastfeeding may protect against obesity.
• BMJ 2007; 335: 815-20.  Longer time of breastfeeding does not reduce allergy/asthma. n=17,046 pairs of mother-infant (13,889 followed up at age 6.5yrs)