I have always loved the idea of biomimicry, even before I knew that anybody else thought about it outside of specific applications. Ever since I was little, when found out that the strongest animal in the world, the rhinceros beetle, could lift 850 times its weight (like a human lifting 65 tons), I've felt like understanding the mechanics of the incredible insect could result in amazing applications for humans in many different ways. We could probably learn something useful from a very large number of species. From Wikipedia:

One example is the attempt to learn from and emulate the incredible ability of termites to maintain virtually constant temperature and humidity in their Sub-Saharan Africa homes despite an outside temperature variation from 3 °C and 42 °C (35 °F at night to 104 °F during the day.) Project TERMES (Termite Emulation of Regulatory Mound Environments by Simulation) scanned a termite mound, created 3-D images of the mound structure and provided the first ever glimpse of construction that may likely change the way we build our own buildings. The Eastgate Centre, a mid-rise office complex in Harare, Zimbabwe, (highlighted in this Biomimicry Institute case-study) stays cool without air conditioning and uses only 10% of the energy of a conventional building its size.
Another example is modeling the echolocation of bats in darkness and adapting that functionality into a cane for the visually impaired. Research performed at the University of Leeds (in the UK) led to the UltraCane, a product manufactured, marketed and sold by Sound Foresight Ltd.

Another popular example is how the beak of the kingfisher bird inspired the nose cone of the Japanese Shinkansen bullet train. The list goes on and on.

Here are some problems I think biomimicry might ultimately solve if the world's economy doesn't completely and irreparably collapse.

• desalination of sea water when not enough clean fresh water is available
• safe transportation not requiring an insane amount of resources (metals to build, etc.)
• refrigeration
• cooling
• heating
• antibiotics
• housing that is safe from natural disasters