Elimination communication claims that parents can mostly if not completely eliminate nappies by instead learning to predict when the baby needs to go and inviting the baby to urinate or defecate by using a cueing sound that the child eventually associates with going to the bathroom as a Pavlovian reaction.
Many native peoples in tropical climates historically raised children without nappies; since the babies were mostly naked anyway the parents would try to anticipate bowel movements and urination and have the baby go in the latrine when possible. This philosophy has been reintroduced in the past few decades under the name elimination communication.
Elimination communication site Diaper-Free Baby points out that this technique removes concerns of nappy rash and most other skin irritations, reduces the risks of urinary tract infections, and also reduces the risk of bed-wetting as the child ages. Some adherents practice full-time elimination communication while others use nappies at night and when out of the home.
It's definitely not for everyone, but elimination communication has a long track record of at least being somewhat effective. The degree to which occasional accidents bothers you will determine if this technique might work for your family.
Practicing Elimination Communication [Diaper-Free Baby]