This paper concerns elliptic curves defined over quadratic fields and having good reduction at all primes. All those real fields admitting such curves having a 2-division point defined over the field and a global minimal model are characterized. The number of isomorphism classes, over the ground field, of these curves is also determined. If the number of divisor classes of the field is odd, all the mentioned curves without a global minimal model are classified and counted as well. It is shown that there are only eight elliptic curves defined over a quadratic field having good reduction everywhere and four 2-division points defined over the field. © 1990 by Pacific Journal of Mathematics.