Normal human tissue is organized into cell lineages, in which the highly differentiated cells that perform tissue functions are the end product of orderly tissue-specific sequences of divisions that start with stem cells or progenitor cells. In this talk we will present a mathematical model of tissue regulation in cell lineages and discuss the type of phenotypic mutations that can lead to an escape from these regulatory mechanisms and eventually to cancer. Next we will explore the role of cellular replication limits in cell lineages. In particular, we will look at how replicative limits can present a barrier to precancerous mutations by influencing the fate of altered but non-neoplastic cells in healthy tissue.