Ideally, your genome does not change at all, from the fertilization of an egg until the day you die. Your epigenome, however -- the collection of methyl groups and so on strapped onto your DNA that promote or inhibit certain genes -- can and does change throughout life.
Cells can acquire changes in their genetic structure, either through epigenetic programming (what Chroot mentioned) or through damage/errors. Processes like these can lead to cancer.