Motivated by the rapid expansion in the development of replication-competent viral agents for the treatment of solid tumors, we formulated and analyzed a three-dimensional mathematical model of a tumor that is infected by a replication-competent virus. We initially considered three patterns of intratumoral injection in which a fixed fraction of cells are initially infected with the virus throughout (a) the entire tumor, (b) the tumor core, and (c) the tumor rim, respectively. For each injection pattern, an approximate analysis of the model provides a simple and accurate condition for whether the virus will eradicate the tumor. The model was then generalized to incorporate nutrient-limited necrosis and an innate immune response against virus-infected tumor cells. Recent preclinical and clinical data were used to validate the model and estimate key parameter values. Our analysis has the following implications: even in the absence of an immune response, tumor eradication requires widespread distribution of the virus within the tumor at the time of infection; core or rim injections alone may result in tumor escape, particularly in a well-vascularized tumor; the more rapidly a virus lyses infected cells the more effective it will be at controlling the tumor; and the innate immune response to the virus can potentially prevent the virus from controlling the tumor, even with repeat injections. Therefore, in addition to diffuse intratumoral infection, tumor eradication by oncolytic adenovirus will probably require potent suppression of innate immune clearance mechanisms (e.g., by replacement of adenovirus E3 genes), combinations with traditional (chemotherapy, radiotherapy) treatments, and/or concomitant therapeutic gene expression with resultant bystander effects.