Using neutron diffraction, we measured the sizes of antiferromagnetic domains in three ferromagnet/ antiferromagnet bilayer samples as a function of the magnitude and sign of exchange bias, temperature, and antiferromagnet composition. Neutron-scattering techniques were applied to thin films with masses less than 10 g. We found the antiferromagnetic domain size to be consistently small regardless of the exchange bias. For a Co/untwinned single crystalline antiferromagnet ͑AF͒-fluoride bilayer, the antiferromagnetic domain size is comparable to the crystallographic domain size of the AF. For one sample the highest temperature at which the exchange bias was nonzero ͑i.e., the blocking temperature͒ was suppressed by ϳ3 K compared to the Néel temperature of the antiferromagnet.