Biological evidence suggests that adaptation of synaptic delays on short to medium timescales plays an important role in learning in the brain. Inspired by biology, we explore the feasibility and power of using synaptic delays to solve challenging tasks even when the synaptic weights are not trained but kept at randomly chosen fixed values. We show that training ONLY the delays in feed-forward spiking networks using backpropagation can achieve performance comparable to the more conventional weight training. Moreover, further constraining the weights to ternary values does not significantly affect the networks' ability to solve the tasks using only the synaptic delays. We demonstrate the task performance of delay-only training on MNIST and Fashion-MNIST datasets in preliminary experiments. This demonstrates a new paradigm for training spiking neural networks and sets the stage for models that can be more efficient than the ones that use weights for computation.