We present algorithms and lower bounds for the Longest Increasing Subsequence (LIS) and Longest Common Subsequence (LCS) problems in the data-streaming model. To decide if the LIS of a given stream of elements drawn from an alphabet \alphabet has length at least $k$, we discuss a one-pass algorithm using $O(k \log \alphabetsize)$ space, with update time either $O(\log k)$ or $O(\log \log \alphabetsize)$; for $\alphabetsize = O(1)$, we can achieve $O(\log k)$ space and constant-time updates. We also prove a lower bound of $\Omega(k)$ on the space requirement for this problem for general alphabets $\alphabet$, even when the input stream is a permutation of $\alphabet$. For finding the actual LIS, we give a $\ceil{\log(1+1/\eps)}$-pass algorithm using $O(k^{1+\eps}\log \alphabetsize)$ space, for any $\eps > 0$. For LCS, there is a trivial $\Theta(1)$-approximate $O(\log n)$-space streaming algorithm when $\alphabetsize = O(1)$. For general alphabets $\alphabet$, the problem is much harder. We prove several lower bounds on the LCS problem, of which the strongest is the following: it is necessary to use $\Omega(n/\rho^2)$ space to approximate the LCS of two $n$-element streams to within a factor of~$\rho$, even if the streams are permutations of each other.