Abstract: This paper is concerned with the study of a non-local Burgers equation for positive bounded periodic initial data. The equation reads u_t – u |D| u + |D|(u^2) = 0. We construct global classical solutions starting from smooth positive data, and global weak solutions starting from data in $L^\infty$. We show that any weak solution is instantaneously regularized into $C^\infty$. We also describe the long-time behavior of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.