We review current challenges in understanding the values and origins of the spins of black holes in binaries. Thanks to recent advances in astrophysical instrumentation, the spins can now be measured using both gravitational waves emitted by merging black holes and electromagnetic radiation from accreting X-ray binaries containing black holes. A key finding of the gravitational-wave observatories is that premerger black holes in binaries have low spin values, with an average dimensionless spin parameter of a, similar to 0.1-0.2, with 90% having a, G 0.6. This implies that the natal spins of black holes are generally low, and the angular momentum transport in massive stars is efficient. On the other hand, most of the published spins in X-ray binaries are very high, a, >= 0.7. In particular, this is the case for binaries with high-mass donors (potential progenitors of mergers), where their published spins range from 0.8 to 1.0. At the same time, their short lifetimes prevent significant spin-up by accretion. Those with low-mass donors could be spun-up to a, >= 0.5 by conservative accretion. Spins a, >= 0.7 can be achieved only if the donor initial masses were more than several solar masses, which remains unproven. However, the existing methods of spin measurements suffer from significant systematic errors. The method relying on relativistic X-ray line broadening is based on the separation of the observed spectra into incident and reflected ones, which is intrinsically highly uncertain. The method relying on spectral fitting of accretion disk continua uses models that predict the disk to be highly unstable, while stability is observed. Improved stable models yield disk temperatures higher than the standard models, and consequently predict lower spins. The published spin measurements in X-ray binaries are uncertain. The spins of the binaries with high-mass donors may be low, while those with low-mass donors have a broader spin distribution, ranging from low to high, including high spins as required to power relativistic jets.