A spectrahedron is an affine slice of the cone of positive semidefinite matrices. Spectrahedra form a rich class of convex bodies that are computationally tractable and appear in many areas of mathematics. Examples include polytopes, ellipsoids, and more exotic convex sets, like the convex hull of some curves. I will introduce the theory of spectrahedra with many examples and discuss some applications in distance geometry and combinatorial optimization.