Let $X $ be a real subvariety of codimension $\ell$ in the complex projective space ${\mathbb P}^d$.

We say that $X$ is hyperbolic with respect to a real linear space $V$ of dimension $\ell-1$

if $X \cap V = \emptyset$ and $X$ intersects any real linear space of dimension $\ell$ through $V$

in real points only.

Alternatively, if $Y$ is the associated hypersurface of $X$ in the Grassmanian ${\mathbb G}(\ell-1,d)$

of $\ell-1$-dimensional linear spaces in ${! \mathbb P}^d$, then $V \not\in Y$ and $Y$ intersects any

real one-dimensional Schubert cycle through $V$ in real points only.

In the case $\ell=1$, i.e., $X$ is a hypersurface, this simply means that $X$ is the zero locus of a homogeneous

hyperbolic polynomial.

I will discuss hyperbolic subvarieties of a higher codimension, the analogues of hyperbolicity cones,

and a class of definite Hermitian determinantal representations that witnesses hyperbolicity.

It turns out that the analogue of the Lax conjecture holds --- any real curve in ${\mathbb P}^d$ that is hyperbolic

with respect to some $d-2$-dimensional linear space admits a definite Hermitian, or even real symmetric, determinantal representation.