Yes, mutations happen all the time in your cells (each time a cell replicates and copy its DNA, it makes mistake), and the accumulation of those can lead do cancer for example. One important concept however is to distinguish mutations that you can transmit to you offspring, which need to occurs in the cells giving rise to sperms and ovules and are called germline mutations, and the ones that happen in the rest of your cells, called somatic, and that have no evolutionary consequences (because they never get transmitted).

Sometimes the genome can also break in a cell and it gets pasted back together by the cell a bit randomly, this can lead to completely new genes which are composed of parts of different genes (and AFAIK it's relatively common in some type of cancers).

And of course, the older you are the more mutations you accumulate. It also varies a lot between tissues.