Relative to the randomized controlled trial (RCT), the basic regression discontinuity (RD) design suffers from lower statistical power and lesser ability to generalize causal estimates away from the treatment eligibility cutoff. This chapter seeks to mitigate these limitations by adding an untreated outcome comparison function that is measured along all or most of the assignment variable. When added to the usual treated and untreated outcomes observed in the basic RD, a comparative RD (CRD) design results. One version of CRD adds a pretest measure of the study outcome (CRD-Pre); another adds posttest outcomes from a nonequivalent comparison group (CRD-CG). We describe how these designs can be used to identify unbiased causal effects away from the cutoff under the assumption that a common, stable functional form describes how untreated outcomes vary with the assignment variable, both in the basic RD and in the added outcomes data (pretests or a comparison group's posttest). We then create the two CRD designs using data from the National Head Start Impact Study, a large-scale RCT. For both designs, we find that all untreated outcome functions are parallel, which lends support to CRD's identifying assumptions. Our results also indicate that CRD-Pre and CRD-CG both yield impact estimates at the cutoff that have a similarly small bias as, but are more precise than, the basic RD's impact estimates. In addition, both CRD designs produce estimates of impacts away from the cutoff that have relatively little bias compared to estimates of the same parameter from the RCT design. This common finding appears to be driven by two different mechanisms. In this instance of CRD-CG, potential untreated outcomes were likely independent of the assignment variable from the start. This was not the case with CRD-Pre. However, fitting a model using the observed pretests and untreated posttests to account for the initial dependence generated an accurate prediction of the missing counterfactual. The result was an unbiased causal estimate away from the cutoff, conditional on this successful prediction of the untreated outcomes of the treated. (author abstract)