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Funding of basic, curiosity-driven science is the lifeblood of scientific and technological innovation. Various models of funding allocation became institutionalized in the 20th century, shaping the present landscape of research funding. There are numerous reasons for scientists to be dissatisfied with current funding schemes, including the imbalance between funding for basic research and mission-directed funding, regional and country disparities, gender disparities, low inter-reviewer reliability, path-dependency of who gets funded, and the trade-off between the effort and time spent on writing or reviewing proposals and doing research. We discuss possible alternative models for dealing with these issues, such as distributed or concentrated funding, placing more weight on the proposals or on the investigators, representative composition of panel members, and retrospective versus prospective awards. We also consider more radical alternatives to current funding schemes, such as partial lotteries in response to low inter-reviewer reliability and the introduction of international competitive applications to a World Research Council alongside national funding sources. There is likely no single best way to fund basic, curiosity-driven research; we examine arguments for and against the possibility of systematically evaluating alternative models empirically.