We have carried out elastic, quasielastic ($\ensuremath{\int} \mathrm{dE}$), and inelastic neutron scattering studies of the antiferromagnetic spin correlations in ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$ with $x$ varying between 0.02 and 0.18. The crystals, which were grown in three different laboratories, exhibit behavior that varies smoothly with $x$. In all cases, antiferromagnetic correlations with a scattering amplitude corresponding to a fully occupied ${\mathrm{Cu}}^{2+}$ square lattice are observed. However, the N\'eel state is destroyed by the doping and the spin-spin correlation length $\ensuremath{\xi}$ is quite short, varying from \ensuremath{\sim}35 to \ensuremath{\sim}8 \AA{} as $x$ varies between 0.02 and 0.18; the local order is, however, the same as in pure ${\mathrm{La}}_{2}$Cu${\mathrm{O}}_{4}$. The fluctuations are dynamic in character as in ${\mathrm{La}}_{2}$Cu${\mathrm{O}}_{4}$ above the N\'eel temperature ${T}_{N}$. To a first approximation, $\ensuremath{\xi}=\frac{3.8}{\sqrt{x}}$ \AA{}, the average separation between the holes introduced by the ${\mathrm{Sr}}^{2+}$ doping. The $x=0.08$ sample exhibits superconductivity with ${T}_{c}=10$ K and with a Meissner fraction exceeding 15% at 5 K; no important differences in the magnetic scattering are observed in the normal and superconducting states. In an appendix we present additional data on the spin dynamics in pure ${\mathrm{La}}_{2}$Cu${\mathrm{O}}_{4}$ at ${T}_{c}=300$ K in a sample with ${T}_{N}=235$ K.