The rapid-neutron capture process ($r$ process) is identified as the producer of about 50% of elements heavier than iron. This process requires an astrophysical environment with an extremely high neutron flux over a short amount of time ($sim$ seconds), creating very neutron-rich nuclei that are subsequently transformed to stable nuclei via $beta^-$ decay. One key ingredient to large-scale $r$-process reaction networks is radiative neutron-capture ($n,gamma$) rates, for which there exist virtually no data for extremely neutron-rich nuclei involved in the $r$ process. Due to the current status of nuclear-reaction theory and our poor understanding of basic nuclear properties such as level densities and average $gamma$-decay strengths, theoretically estimated ($n,gamma$) rates may vary by orders of magnitude and represent a major source of uncertainty in any nuclear-reaction network calculation of $r$-process abundances. In this review, we discuss new approaches to provide information on neutron-capture cross sections and reaction rates relevant to the $r$ process. In particular, we focus on indirect, experimental techniques to measure radiative neutron-capture rates. While direct measurements are not available at present, but could possibly be realized in the future, the indirect approaches present a first step towards constraining neutron-capture rates of importance to the $r$ process.
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