By means of a variational approach we rigorously deduce three one-dimensional
models for elastic ribbons from the theory of von K\'arm\'an plates, passing to
the limit as the width of the plate goes to zero. The one-dimensional model
found starting from the "linearized" von K\'arm\'an energy corresponds to that
of a linearly elastic beam that can twist but can deform in just one plane;
while the model found from the von K\'arm\'an energy is a non-linear model that
comprises stretching, bendings, and twisting. The "constrained" von K\'arm\'an
energy, instead, leads to a new Sadowsky type of model.