  A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge
set $E\subseteq 2^V$; the edges in $E$ are not required to all have the same
cardinality. The set of all cardinalities of edges in $H$ is denoted by $R(H)$,
the set of edge types. For a fixed hypergraph $H$, the Tur\'an density $\pi(H)$
is defined to be $\lim_{n\to\infty}\max_{G_n}h_n(G_n)$, where the maximum is
taken over all $H$-free hypergraphs $G_n$ on $n$ vertices satisfying
$R(G_n)\subseteq R(H)$, and $h_n(G_n)$, the so called Lubell function, is the
expected number of edges in $G_n$ hit by a random full chain. This concept,
which generalizes the Tur\'an density of $k$-uniform hypergraphs, is motivated
by recent work on extremal poset problems. The details connecting these two
areas will be revealed in the end of this paper.
  Several properties of Tur\'an density, such as supersaturation, blow-up, and
suspension, are generalized from uniform hypergraphs to non-uniform
hypergraphs. Other questions such as "Which hypergraphs are degenerate?" are
more complicated and don't appear to generalize well. In addition, we
completely determine the Tur\'an densities of ${1,2}$-hypergraphs.
