  We used the Atacama Large Millimeter/submillimeter Array (ALMA), covering a
nearly contiguous 289 GHz frequency range between 84.2 and 373.2 GHz, to image
the continuum and spectral line emission at 1.6\arcsec ($\sim 28$ pc)
resolution down to a sensitivity of $30-50$ mK. This article describes the ALMA
Comprehensive High-resolution Extragalactic Molecular Inventory (ALCHEMI) Large
Program. We focus on the analysis of the spectra extracted from the $15''$
($\sim255$ pc) resolution ALMA Compact Array data. We model the molecular
emission assuming local thermodynamic equilibrium with 78 species detected.
Additionally, multiple hydrogen and helium recombination lines are identified.
Spectral lines contribute 5 to 36\% of the total emission in frequency bins of
50 GHz. We report the first extragalactic detections of C$_2$H$_5$OH, HOCN,
HC$_3$HO, and several rare isotopologues. Isotopic ratios of carbon, oxygen,
sulfur, nitrogen and silicon were measure with multiple species. Infrared
pumped vibrationaly excited HCN, HNC, and HC$_3$N emission, originating in
massive star formation locations, is clearly detected at low resolution, while
we do not detect it for HCO$^+$. We suggest high temperature conditions in
these regions driving a seemingly "carbon-rich" chemistry which may also
explain the observed high abundance of organic species close to those in
Galactic hot cores. The $L_{vib}/L_{IR}$ ratio is used as a proxy to estimate a
$3\%$ contribution from proto super star cluster to the global infrared
emission. Measured isotopic ratios with high dipole moment species agree with
those within the central kiloparsec of the Galaxy, while those derived from
$\rm^{13}C^{18}O$ are a factor of 5 larger, confirming the existence of
multiple ISM components within NGC 253 with different degrees of
nucleosynthesis enrichment. ALCHEMI provides a template for early Universe
galaxies.
