  Understanding the electron pairing in hole-doped cuprate superconductors has
been a challenge, in particular because the "normal" state from which it
evolves is unprecedented. Now, after three and a half decades of research,
involving a wide range of experimental characterizations, it is possible to
delineate a clear and consistent cuprate story. It starts with doping holes
into a charge-transfer insulator, resulting in in-gap states. These states
exhibit a pseudogap resulting from the competition between antiferromagnetic
superexchange $J$ between nearest-neighbor Cu atoms (a real-space interaction)
and the kinetic energy of the doped holes, which, in the absence of
interactions, would lead to extended Bloch-wave states whose occupancy is
characterized in reciprocal space. To develop some degree of coherence on
cooling, the spin and charge correlations must self-organize in a cooperative
fashion. A specific example of resulting emergent order is that of spin and
charge stripes, as observed in La$_{2-x}$Ba$_x$CuO$_4$. While stripe order
frustrates bulk superconductivity, it nevertheless develops pairing and
superconducting order of an unusual character. The antiphase order of the spin
stripes decouples them from the charge stripes, which can be viewed as
hole-doped, two-leg, spin-$\frac12$ ladders. To achieve superconducting order,
the pair correlations in neighboring ladders must develop phase order. In the
presence of spin stripe order, antiphase Josephson coupling can lead to
pair-density-wave superconductivity. Alternatively, in-phase superconductivity
requires that the spin stripes have an energy gap, which empirically limits the
coherent superconducting gap. Hence, superconducting order in the cuprates
involves a compromise between the pairing scale, which is maximized at
$x\sim\frac18$, and phase coherence, which is optimized at $x\sim0.2$.
