The two major functions of a scientific publishing system are to provide access to and evaluation of scientific papers. While open access (OA) is becoming a reality, open evaluation (OE), the other side of the coin, has received less attention. Evaluation steers the attention of the scientific community and thus the very course of science. It also influences the use of scientific findings in public policy. The current system of scientific publishing provides only journal prestige as an indication of the quality of new papers and relies on a non-transparent and noisy pre-publication peer-review process, which delays publication by many months on average. Here I propose an OE system, in which papers are evaluated post-publication in an ongoing fashion by means of open peer review and rating. Through signed ratings and reviews, scientists steer the attention of their field and build their reputation. Reviewers are motivated to be objective, because low-quality or self-serving signed evaluations will negatively impact their reputation. A core feature of this proposal is a division of powers between the accumulation of evaluative evidence and the analysis of this evidence by paper evaluation functions (PEFs). PEFs can be freely defined by individuals or groups (e.g., scientific societies) and provide a plurality of perspectives on the scientific literature. Simple PEFs will use averages of ratings, weighting reviewers (e.g., by H-index), and rating scales (e.g., by relevance to a decision process) in different ways. Complex PEFs will use advanced statistical techniques to infer the quality of a paper. Papers with initially promising ratings will be more deeply evaluated. The continual refinement of PEFs in response to attempts by individuals to influence evaluations in their own favor will make the system ungameable. OA and OE together have the power to revolutionize scientific publishing and usher in a new culture of transparency, constructive criticism, and collaboration.