We present reverberation mapping results for the MgII{lambda}2800{AA} broad emission line in a sample of 193 quasars at 0.35<z<1.7 with photometric and spectroscopic monitoring observations from the Sloan Digital Sky Survey Reverberation Mapping project during 2014-2017. We find significant time lags between the MgII and continuum lightcurves for 57 quasars, and define a "gold sample" of 24 quasars with the most reliable lag measurements. We estimate false-positive rates for each lag that range from 1% to 24%, with an average false-positive rate of 11% for the full sample and 8% for the gold sample. There are an additional ~40 quasars with marginal MgII lag detections, which may yield reliable lags after additional years of monitoring. The MgII lags follow a radius-luminosity relation with a best-fit slope that is consistent with {alpha}=0.5, but with an intrinsic scatter of 0.36dex that is significantly larger than found for the H{beta} radius-luminosity relation. For targets with SDSS-RM lag measurements of other emission lines, we find that our MgII lags are similar to the H{beta} lags and ~2-3 times larger than the CIV lags. This work significantly increases the number of MgII broad-line lags and provides additional reverberation-mapped black hole masses, filling the redshift gap at the peak of supermassive black hole growth between the H{beta} and CIV emission lines in optical spectroscopy.