Over the past decade, DNA-based data storage systems have emerged as a promising solution to address the ever-increasing volume of information, due to their density, durability, and low maintenance costs. The main difference between retrieving data from DNA and traditional storage media is that DNA strands are read randomly until the desired information is obtained. The coverage depth problem concerns the scenario where all data must be recovered; in the language of coding theory, this corresponds to considering a generator matrix G of an [n,k]_q code C and computing the expected number of columns that must be drawn until the entire space F_q^k is spanned. Assuming that the columns are drawn uniformly at random and the channel is noiseless, MDS codes are optimal. However, MDS codes do not exist over small fields, raising the question of which alternative codes offer the best performance. This talk provides an overview of the problem of retrieving data from DNA and presents new results on the performance of various code families over small alphabets in this context. The techniques employed range from probability to duality theory and combinatorics.