Large-scale distributed storage systems typically use erasure codes to provide durability of data in the face of failures. A set of $k$ blocks to be stored is encoded using an $[n, k]$ code to generate $n$ blocks that are then stored on different storage nodes. The redundancy configuration is chosen based on the failure rates of storage devices, and is typically kept constant. However, a recent work by Kadekodi et al. shows that the failure rate of storage devices vary significantly over time, and that adapting the redundancy configuration in response to such variations provides significant benefits. Converting the redundancy configuration of already encoded data by re-encoding requires significant overhead on resources such as accesses, device IO, network bandwidth, and compute cycles. In this work, we first present a framework to formalize the notion of code conversion: the process of converting data encoded with an $[n^I, k^I]$ code into data encoded with an $[n^F, k^F]$ code while maintaining desired decodability properties, such as the maximum-distance-separable (MDS) property. We then introduce convertible codes, a new class of codes that allow for code conversions in a resource-efficient manner. For an important parameter regime (which we call the merge regime) along with the linearity and MDS decodability constraint, we prove tight bounds on the number of nodes accessed during code conversion. In particular, our achievability result is an explicit construction of MDS convertible codes that are optimal for all parameter values in the merge regime albeit with a high field size. We then present explicit low-field-size constructions of optimal MDS convertible codes for a broad range of parameters in the merge regime. Our results thus show that it is indeed possible to achieve code conversions with significantly lesser resources as compared to re-encoding.