  The positive Grassmannian $Gr_{k,n}^{\geq 0}$ is the subset of the real
Grassmannian where all Pl\"ucker coordinates are nonnegative. It has a
beautiful combinatorial structure as well as connections to statistical
physics, integrable systems, and scattering amplitudes. The amplituhedron
$\mathcal{A}_{n,k,m}(Z)$ is the image of the positive Grassmannian
$Gr_{k,n}^{\geq 0}$ under a positive linear map $\mathbb{R}^n \to
\mathbb{R}^{k+m}$. We will explain how ideas from oriented matroids, tropical
geometry, and cluster algebras shed light on the structure of the positive
Grassmannian and the amplituhedron.
