Gradient clipping is a widely-used technique in the training of deep networks, and is generally motivated from an optimisation lens: informally, it controls the dynamics of iterates, thus enhancing the rate of convergence to a local minimum. This intuition has been made precise in a line of recent works, which show that suitable clipping can yield significantly faster convergence than vanilla gradient descent. In this paper, we study gradient clipping from an robustness lens: informally, one expects clipping to provide robustness to noise, since one does not overly trust any single sample. Surprisingly, we prove that gradient clipping does not in general provide robustness to label noise. On the other hand, we show that robustness is achieved by a form of loss clipping. This yields a simple, noise-robust alternative to the standard cross-entropy loss which performs well empirically.