Although higher structures have been around for quite some time, they recently have come back into focus through renewed interest in higher categories. There are several reasons for this.
In geometry one is trying to interpret extended cobordism theories, where the higher structures are meant to mimic higher codimensions. An analogue in algebra is known to the 2-categorical level, the prime example being the 2-category of rings, bi-modules and bi-module morphisms. Beyond this there are many open questions of fundamental nature. The central problem is what type of coherence to require.
In physics higher structures naturally appear in two related fashions. The first is through the extended field theories and the second through field theories with defects. This is mathematically mimicked by cobordisms and defect lines and points abstractly interpreted as inclusions into higher dimensional objects.
The "truncated" versions of higher structures can be assembled into infinity up to a homotopy everything version. This is the setting of the influential program of Lurie which provides firm foundations to derived algebraic geometry, and, hopefully, to higher differential geometry which is not yet that well established.
Geometric and physical points of view combine in the constructions of string topology and in the proofs of the cobordism hypothesis. One approach to this, which will also be an integral part of this program, is the operadic/monadic point of view as many liigher categorical structures can be interpreted as actions of certain liigher dimensional operads/monads. The classical homotopy theory teaches us that this is the correct way to encode higher homotopies and homotopical algebra in general.
The complexity of higher dimensional structures and necessity to work with them efficiently has required reconsideration of the foundations of mathematics. A new theory called univalent foundations or homotopy type theory emerges in recent years which has a potential to become a common language for mathematicians working with higher categorical structures. We wish to include this theory as a supplement to our main topics, but also as a possible future direction of research.