Three major cell populations in the body don't undergo mitosis. Post-mitotic cells don't divide further after they reach maturity. These include erythrocytes, platelets, neurons, retinal photoreceptors, cardiac cells and skeletal muscle cells. Quiescent cell populations aren't actively progressing through the cell cycle but are still capable of mitosis if stimulated. Finally, senescent cells damaged by wear and tear don't undergo mitosis.
The reason mature red blood cells and platelets don't divide is that they lack nuclei. Erythrocytes lose their nuclei at the reticulocyte stage of development. Platelets are cytosolic fragments of giant bone cells called megakaryocytes; hence, they were never nucleated in the first place.
As a general rule, neurons don't undergo mitosis except for the progenitors of taste bud cells and possibly a few neurons in the brain's hippocampus. Glial cells such as astrocytes and oligodendrocytes share a common embryonic origin with neurons; however, these cells can proliferate, especially in response to trauma.
Skeletal and cardiac muscle fibers present a puzzle. Skeletal muscle cells have multiple nuclei but undergo little or no mitosis after fetal life. Similarly, cardiac cells spend their lives in a stage of the cell cycle known as G2 arrest. Although this prevents cardiac cells from regenerating after a heart attack, it also accounts for the rare incidence of tumors arising from skeletal or cardiac muscle.