  Complex systems are fascinating because their rich macroscopic properties
emerge from the interaction of many simple parts. Understanding the building
principles of these emergent phenomena in nature requires assessing natural
complex systems experimentally. However, despite the development of large-scale
data-acquisition techniques, experimental observations are often limited to a
tiny fraction of the system. This spatial subsampling is particularly severe in
neuroscience, where only a tiny fraction of millions or even billions of
neurons can be individually recorded. Spatial subsampling may lead to
significant systematic biases when inferring the collective properties of the
entire system naively from a subsampled part. To overcome such biases, powerful
mathematical tools have been developed in the past. In this perspective, we
overview some issues arising from subsampling and review recently developed
approaches to tackle the subsampling problem. These approaches enable one to
assess, e.g., graph structures, collective dynamics of animals, neural network
activity, or the spread of disease correctly from observing only a tiny
fraction of the system. However, our current approaches are still far from
having solved the subsampling problem in general, and hence we conclude by
outlining what we believe are the main open challenges. Solving these
challenges alongside the development of large-scale recording techniques will
enable further fundamental insights into the working of complex and living
systems.
