  A cubic space is a vector space equipped with a symmetric trilinear form.
Using categorical Fra\"iss\'e theory, we show that there is a universal
ultrahomogeneous cubic space $V$ of countable infinite dimension, which is
unique up to isomorphism. The automorphism group $G$ of $V$ is quite large and,
in some respects, similar to the infinite orthogonal group. We show that $G$ is
a linear-oligomorphic group (a class of groups we introduce), and we determine
the algebraic representation theory of $G$. We also establish some
model-theoretic results about $V$: it is $\omega$-categorical (in a modified
sense), and has quantifier elimination (for vectors). Our results are not
specific to cubic spaces, and hold for a very general class of tensor spaces;
we view these spaces as linear analogs of the relational structures studied in
model theory.
