We measure the low-temperature resistance of permalloy break junctions as a function of contact size and the magnetic field angle, in applied fields large enough to saturate the magnetization. For both nanometer-scale metallic contacts and tunneling devices we observe large changes in resistance with angle, as large as 25% in the tunneling regime. The pattern of magnetoresistance is sensitive to changes in bias on a scale of a few mV. We interpret the effect as a consequence of conductance fluctuations due to quantum interference. PACS numbers: 72.25.Ba; 73.63.Rt; 75.75.+a The magnetoresistance properties of nanometer-scale magnetic devices can be quite different from those of larger samples. One aspect of this difference has been explored extensively in previous experiments -the resistance of magnetic domain walls created when the magnetic moment direction in one magnetic electrode is rotated relative to the moment in a second electrode [1,2,3,4,5,6,7,8,9]. Here we focus on a different aspect of the physics of magnetoresistance in nanoscale magnetic contacts -the anisotropic magnetoresistance (AMR) that arises when the magnetization throughout a device is rotated uniformly so as to change the angle between the direction of current flow and the magnetic moment. Our measurements are motivated by predictions of increased AMR for atomic-sized ballistic conductors [10] and indications of enhanced AMR in Ni contacts [8]. By making detailed studies of resistance as a function of field angle using mechanically-stable permalloy contacts, we show that the size of the AMR signal at low temperature can increase dramatically as the contact cross section is narrowed to the nanometer-scale regime. Even more strikingly, we find that point contacts which are completely broken, so as to enter the tunneling regime, also exhibit a tunneling anisotropic magnetoresistance effect (TAMR) as large as 25% when the magnetic-moment directions in the two contacts are rotated together while remaining parallel.Magnetostriction and magnetostatic forces can alter the geometry of nanoscale junctions as the magnetic field is varied, and produce artifacts in the resistance, so experiments must be designed to minimize these effects [5,6,7]. For this reason, our contacts are firmly attached to a non-magnetic silicon substrate and are measured entirely at low temperature to suppress thermally-driven surface diffusion of metal atoms. Similar structures have proven [8,9] to be much more mechanically-stable than previous samples which were measured at room temperature. We fabricate our devices using aligned steps of electron beam lithography to first pattern 20-nm-thick gold contact pads and then 30-nm-thick magnetic permalloy (Py = Ni 80 Fe 20 ) point contacts [9]. Each contact con- sists of two elongated electrodes which are connected by a 100-nm-wide bridge ( Fig. 1(b)). The magnetic field B is applied using a 3-coil vector magnet capable of 0.9 T in any direction and up to 7 T along one axis (the x axis, defined below) with the other two coils t...