Magnetic interactions are generally believed to play a key role in mediating electron pairing for superconductivity in iron arsenides; yet their character is only partially understood. Experimentally, the antiferromagnetic (AF) transition is always preceded by or coincident with a tetragonal to orthorhombic structural distortion. Although it has been suggested that this lattice distortion is driven by an electronic nematic phase, where a spontaneously generated electronic liquid crystal state breaks the C4 rotational symmetry of the paramagnetic state, experimental evidence for electronic anisotropy has been either in the low-temperature orthorhombic phase or the tetragonal phase under uniaxial pressure that breaks this symmetry. Here we use inelastic neutron scattering to demonstrate the presence of a large in-plane spin anisotropy above TN in the unstressed tetragonal phase of BaFe2As2. In the low-temperature orthorhombic phase, we find highly anisotropic spin waves with a large damping along the AF a-axis direction. On warming the system to the paramagnetic tetragonal phase, the low-energy spin waves evolve into quasi-elastic excitations, while the anisotropic spin excitations near the zone boundary persist. These results strongly suggest that the spin nematicity we find in the tetragonal phase of BaFe2As2 is the source of the electronic and orbital anisotropy observed above TN by other probes, and has profound consequences for the physics of these materials. ; Comment: paper 20 pages, 4 figures, and supplementary info 11 pages, 7 figures