Abstract We present the first results from the Mapping Obscuration to Reionization with ALMA (MORA) survey, the largest Atacama Large Millimeter/submillimeter Array (ALMA) blank-field contiguous survey to date (184 arcmin 2 ) and the only at 2 mm to search for dusty star-forming galaxies (DSFGs). We use the 13 sources detected above 5 σ to estimate the first ALMA galaxy number counts at this wavelength. These number counts are then combined with the state-of-the-art galaxy number counts at 1.2 and 3 mm and with a backward evolution model to place constraints on the evolution of the IR luminosity function and dust-obscured star formation in the past 13 billion years. Our results suggest a steep redshift evolution on the space density of DSFGs and confirm the flattening of the IR luminosity function at faint luminosities, with a slope of . We conclude that the dust-obscured component, which peaks at z ≈ 2–2.5, has dominated the cosmic history of star formation for the past ∼12 billion years, back to z ∼ 4. At z = 5, the dust-obscured star formation is estimated to be ∼35% of the total star formation rate density and decreases to 25%–20% at z = 6–7, implying a minor contribution of dust-enshrouded star formation in the first billion years of the universe. With the dust-obscured star formation history constrained up to the end of the epoch of reionization, our results provide a benchmark to test galaxy formation models, to study the galaxy mass assembly history, and to understand the dust and metal enrichment of the universe at early times.