  A gaussoid is a combinatorial structure that encodes independence in
probability and statistics, just like matroids encode independence in linear
algebra. The gaussoid axioms of Lnenicka and Mat\'us are equivalent to
compatibility with certain quadratic relations among principal and
almost-principal minors of a symmetric matrix. We develop the geometric theory
of gaussoids, based on the Lagrangian Grassmannian and its symmetries. We
introduce oriented gaussoids and valuated gaussoids, thus connecting to real
and tropical geometry. We classify small realizable and non-realizable
gaussoids. Positive gaussoids are as nice as positroids: they are all
realizable via graphical models.
