  Light is generally expected to travel through isotropic media independent of
its direction. This makes it challenging to develop non-reciprocal optical
elements like optical diodes or circulators, which currently rely on
magneto-optical effects and birefringent materials. Here we present
measurements of non-reciprocal transmission and spontaneous symmetry breaking
between counter-propagating light in dielectric microresonators. The symmetry
breaking corresponds to a resonance frequency splitting that allows only one of
two counter-propagating (but otherwise identical) light waves to circulate in
the resonator. Equivalently, the symmetry breaking can be seen as the collapse
of standing waves and transition to travelling waves within the resonator. We
present theoretical calculations to show that the symmetry breaking is induced
by Kerr-nonlinearity-mediated interaction between the counter-propagating
light. This effect is expected to take place in any dielectric ring-resonator
and might constitute one of the most fundamental ways to induce optical
non-reciprocity. Our findings pave the way for a variety of applications
including all optical switching, nonlinear-enhanced rotation sensing, optically
controllable circulators and isolators, optical flip-flops for photonic
memories as well as exceptionally sensitive power and refractive index sensors.
