In iaido (art of classical Japanese sword-drawing), you start training with a bokken (wooden replica) for at least the first six months. Thereafter, you progress onto an iai-to, (training 'katana'). They are usually alloy-based and unsharpened for safe practise.
In kendo, a shinai is used, which is a 'sword' made of bamboo. In kendo, they mainly practise sparring, hence the bamboo 'sword'.
In other sword-arts, eg. kung fu and tai chi etc, replica wooden and metal (unsharpened) swords are also used.
Have a look at these sites for inspiration: