No, asthma itself is not hereditary, but there does seem to be a hereditary component to the tendency to develop asthma. Although there is some disagreement as to the exact numbers, most researchers agree that the chances climb dramatically if one or both parents have asthma. The actual condition of asthma seems not to be inherited, but the tendency to develop it appears to be. Whether or not an individual develops asthma is also influenced by their exposure to various other factors such as infections, irritants, and allergens. Even a tiny bit of airway narrowing makes it more difficult for a young child to move air in and out of his lungs.
Air moving through narrowed airways causes wheezing and produces the coughing of asthma.
People do not really outgrow asthma, though as they get older their airways may become less sensitive. As the airways grow, small amounts of narrowing have less of an effect. Also, people with asthma learn to avoid triggers, such as cigarette smoke and other irritants. In general, asthma in children gets better as the child gets older. Children with severe asthma are less likely to outgrow it.