Brainz.org has a very good article which gives 15 excellent examples of designs inspired by nature (biomimicry or biomimetics).
READ The 15 Coolest Cases of Biomimicry
The list includes the classic examples of Velcro (adhesion) , air-conditioning (termites) and gecko (adhesion) but also a number of new ones including:
- Insect-inspired Robots
- Friction-reducing sharkskin (e.g. for competitive swimming)
- Morphing aircraft wings which change shape
Interestingly it does not mention any examples of organizational biomimicry which is what inspired bioteams.