Abstract: Let $[n]=\{ 1,2,\ldots , n\}$ and let $\mathcal{F} \subset 2^{[n]}$ be a family of its subsets. Sperner proved in 1928 that if $\mathcal{F}$ contains no pair of members in inclusion that is $F_1, F_2\in \mathcal{F}$ implies $F_1\not\subset F_2$ then $|\mathcal{F}|\leq \binom{n}{\lfloor n/2\rfloor}$. Inspired by an application, Erdős generalized this theorem in the following way. Suppose that $\mathcal{F}$ contains no $k+1$ distinct members $F_1\subset F_2\subset \ldots \subset F_{k+1}$ then $|\mathcal{F}|$ is at most the sum of the $k$ largest binomial coefficients of order $n$. This bound is, of course, tight. We will survey results of this type: determine the largest family of subsets of $[n]$ under a certain condition forbidding a given configuration described solely by inclusion among the members. This maximum is denoted by La$(n,P)$ where $P$ is the forbidden configuration. The final (hopelessly difficult) conjecture is that La$(n,P)$ is asymptotically equal to the sum of the $k$ largest binomial coefficients where the $k$ largest levels contain no configuration $P$, but $k+1$ levels do. Another related problem is the following one. A copy of the given poset $P$ is a family $\mathcal{F}$ of subsets of $[n]$ where the embedding $f: P \rightarrow \mathcal{F}$ maps comparable elements of $P$ into comparable subsets. Two copies $\mathcal{F}_1, \mathcal{F}_2$ of $P$ are incomparable if no member of $\mathcal{F}_1$ is a subset of a member of $\mathcal{F}_2$ and no member of $\mathcal{F}_2$ is a subset of a member of $\mathcal{F}_1$. The maximum number of incomparable copies of $P$ is denoted by LA$(n.P)$. This quantity is asymptotically determined for all $P$, unlike La$(n,P)$.