In this talk I will present results on a stochastic model of spatial evolution on a lattice, motivated by the process of carcinogenesis (or cancer initiation) from healthy epithelial tissue. Cancer often arises through a sequence of genetic alterations or mutations. Each of these alterations may confer a fitness advantage to the cell, resulting in a clonal expansion. To model this we will consider a generalization of the biased voter process which incorporates successive mutations modulating fitness. Under this model we will investigate a possible mechanism for the phenomenon of ``field cancerization," which refers to the clinical observation that multiple independent primary tumors often arise in the same region of tissue. (joint work w/ K. Leder, M. Ryser, and R. Durrett)