We present the design, rationale, properties and catalogs of the MusE Gas FLOw and Wind survey (MEGAFLOW), a survey of the cool gaseous halos of z ≃ 1.0 galaxies using low-ionization Mg ii absorption systems. The survey consists of 22 quasar fields selected from the Sloan Digital Sky Survey (SDSS) having multiple (≥ 3) strong Mg ii absorption lines over the redshift range 0.3 < z < 1.5. Each quasar was observed with the Multi-Unit Spectroscopic Explorer (MUSE) and the Ultraviolet and Visual Echelle Spectrograph (UVES), for a total of 85 hr and 63 hr, respectively. The UVES data resulted in 127 Mg ii absorption lines over 0.25 < z < 1.6, with a median rest-frame equivalent width (REW) 3σ limit of ≈ 0.05 Å. The MUSE data resulted in ∼2400 galaxies of which 1403 with redshift confidence ZCONF> 1, i.e. more than 60 galaxies per arcmin 2 . They were identified using a dual detection algorithm based on both continuum and emission line objects. The achieved [O ii] 50% completeness is 3.7 +0.8 -0.6 × 10 -18 erg s -1 cm -2 (corresponding to SFR> 0.01 M⊙ yr -1 at z = 1) using realistic mock [O ii] emitters and the 50% completeness is mF 775W ≈ 26 AB magnitudes for continuum sources. We find that (i) the fraction of [O ii] emitting galaxies which have no continuum is ∼ 15%; (ii) the success rate in identifying at least one galaxy within 500 km s -1 and 100 kpc is ≈ 90% for Mg ii absorptions with W 2796 r ≳ 0.5 Å ; (iii) the mean number of galaxies per Mg ii absorption is 2.9 ± 1.6 within the MUSE field-of-view; (iv) of the 80 Mg ii systems at 0.3 < z < 1.5, 40 (20) have 1 (2) galaxies within 100 kpc, respectively; (v) all but two host galaxies have stellar masses M⋆ > 10 9 M⊙, and star-formation rates > 1 M⊙ yr -1 .