The Advanced Scientific Computing Research (ASCR) program of the Office of Science (SC), U.S. Department of Energy (DOE), hereby invites proposals for basic mathematical, statistical and computational research that significantly advances uncertainty quantification methodologies for enabling extreme-scale science. The purpose of this FOA is to invite proposals in ASCR Applied Mathematics basic research that significantly advance uncertainty quantification (UQ) methodologies as an enabling technology in extreme-scale scientific computing. UQ broadly refers to the end-to-end study of the accuracy, reliability, development and effective use of computational models in making scientific inferences. Mathematically rigorous UQ methodologies are essential to a wide range of DOE science and engineering applications in carrying out predictions, design optimization, decision making, or other high-level tasks. UQ relies on a broad range of applied mathematics and statistics research, along with algorithmic and computational developments, and subject matter expertise, to enable an appropriate level of confidence in the use of computational models for scientific investigations. A companion Program Announcement to DOE Laboratories (LAB 13-895) will be posted on the SC Grants and Contracts web site at: http://www.science.doe.gov/grants The full text of the Funding Opportunity Announcement (FOA) is located on FedConnect. Instructions for completing the Grant Application Package are contained in the full text of the FOA which can be obtained at: https://www.fedconnect.net/FedConnect/?doc=DE-FOA-0000895&agency=DOE.