Mathematical modeling occupies an unusual space in the undergraduate mathematics curriculum: typically an "advanced" course, it nonetheless has little to do with formal proof, the usual hallmark of advanced mathematics. Mathematics departments are thus forced to decide what role they want the modeling course to play, both as a component of the mathematics major and as a course with potential cross-disciplinary appeal. This article takes the position that a modeling course de-emphasizing mathematical formalism in favor of scientific computing, data analysis, and communication skills is both an appropriate upper-level mathematics course and an effective means of training real-world problem solvers. Designing a course that serves a broad range of students while remaining practically relevant and mathematically sophisticated can be a challenge, however. Here the author distills key features of his recent course with a group of 20 undergraduates from widely different majors. Comments about course outcomes are contextualized by a broader discussion about real-world modeling practices. The piece concludes with a list of concrete ideas about target competencies and the ways in which they can be achieved.