Real world uses of homotopy theory

I covered homotopy theory in a recent maths course. However I was never presented with any reasons as to why (or even if) it is useful.

Is there any good examples of its use outside academia?

Homotopy theory / algebraic topology was born out of applications rather than abstract nonsense considerations. So there's plenty of applications, as that's how the subject began.

Perhaps the first topological proof would be the bridges of Konigsberg problem: http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg

Where algebraic topology started getting off the ground was in the work of Poincare. The Poincare-Hopf Index theorem: http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Hopf_theorem

was a landmark. In its natural setting it was a relationship between Euler characteristic, tangent bundles and intersection theory. But from the perspective of a differential equator it's a fundamental tool that allows you to determine whether or not differential equations have fixed points.

Applications have piled-up over the years. Some of the more modern ones are listed in other people's responses. The birth of topological dynamics in the mid 20-th century was of course a big one.

Robert Ghrist is an amazing applied mathematician who uses a lot of interesting algebraic topology for engineering applications. He uses homology and sheaf theory. I claim this answers your question since homology is a generalization of homotopy theory.

Relevant link: http://www.math.uiuc.edu/~ghrist/index_files/research.htm

Homotopy groups and homotopy classes of maps are used in physics to study topological defects. Roughly speaking, $\pi_k(X)$ classifies codimension $k+1$ defects in textures modeled on $X$ (maps from $R^d$ to $X$ continuous except at the loci of defects). Other homotopy classes of maps can be used to study linked defects.

See the following beautiful review paper by N.D. Mermin for an introduction: http://rmp.aps.org/abstract/RMP/v51/i3/p591_1