Natural systems provide both puzzles to unravel and demonstrations of what's possible. The natural world is full of complex systems of dynamically interchangeable, individually unreliable components that produce effective and reliable outcomes at the group level. A complementary goal to understanding the operation of such systems is that of being able to engineer artifacts that work in a similar way. One notable type of collective behavior is collective construction, epitomized by mound-building termites, which build towering, intricate mounds through the joint activity of millions of independent and limited insects. The artificial counterpart would be swarms of robots designed to build human-relevant structures. I will discuss work on both aspects of the problem, including studies of cues that individual termite workers use to help direct their actions and coordinate colony activity, and development of robot systems that build user-specified structures despite limited information and unpredictable variability in the process. These examples illustrate principles used by the insects and show how they can be applied in systems we create.