We present calculations of the electronic structure of various atoms and molecules in strong magnetic fields ranging from $B={10}^{12}\phantom{\rule{0.3em}{0ex}}\mathrm{G}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}2\ifmmode\times\else\texttimes\fi{}{10}^{15}\phantom{\rule{0.3em}{0ex}}\mathrm{G}$, appropriate for radio pulsars and magnetars. For these field strengths, the magnetic forces on the electrons dominate over the Coulomb forces, and to a good approximation the electrons are confined to the ground Landau level. Our calculations are based on the density functional theory, and use a local magnetic exchange-correlation function which is tested to be reliable in the strong field regime. Numerical results of the ground-state energies are given for ${\mathrm{H}}_{N}$ (up to $N=10$), ${\mathrm{He}}_{N}$ (up to $N=8$), ${\mathrm{C}}_{N}$ (up to $N=5$), and ${\mathrm{Fe}}_{N}$ (up to $N=3$), as well as for various ionized atoms. Fitting formulae for the $B$ dependence of the energies are also given. In general, as $N$ increases, the binding energy per atom in a molecule, $\ensuremath{\mid}{E}_{N}\ensuremath{\mid}∕N$, increases and approaches a constant value. For all the field strengths considered in this paper, hydrogen, helium, and carbon molecules are found to be bound relative to individual atoms (although for $B$ less than a few $\ifmmode\times\else\texttimes\fi{}{10}^{12}\phantom{\rule{0.3em}{0ex}}\mathrm{G}$, carbon molecules are very weakly bound relative to individual atoms). Iron molecules are not bound at $B\ensuremath{\lesssim}{10}^{13}\phantom{\rule{0.3em}{0ex}}\mathrm{G}$, but become energetically more favorable than individual atoms at larger field strengths.