Author Summary Cancer is a disease of multicellular organisms that is characterized by a breakdown of cooperation between individual cells. The progression of cancer proceeds from a single genetically altered cell to billions of invasive cells through a series of clonal expansions. During tumorigenesis the cancer cells undergo replication and mutation, thereby increasing the size and invasiveness of the tumor. Recent sequencing projects of cancer cells suggest that mutations in up to 20 different genes might be responsible for driving an individual tumor's development. This insight contrasts with most mathematical models of cancer progression, which assume that the cancer phenotype is driven by mutations in only a few genes. We present a new mathematical model in which tumorigenesis is driven by mutations in many genes, most of which confer only a small selective advantage. Specifically, the progression of a benign tumor of the colon (adenoma) to a malignant tumor (carcinoma) is described by a Wright-Fisher process with growing population size. We explore the basic parameters of the model that are consistent with observed data. We also derive an analytical formula for the expected waiting time for the progression from benign to maligant tumor in terms of the population size, the mutation rate, the selective advantage, and the number of susceptible genes.