Operads are classical objects in algebraic topology originally used to classify
loop spaces. Since then, they have found natural applications ranging from
mathematical physics to algebraic geometry. In this talk we will define an
important class of these objects, the little $n$-cubes operads, and explain some
current work and applications ranging from understanding moduli spaces of curves
to modelling embeddings of spaces. Material in this talk will include joint work
with Pedro Boavida de Brito, Geoffroy Horel, Philip Hackney and Donald Yau.