Blockchain and distributed ledger technology started as a decentralized infrastructure to enable and manage digital currency like Bitcoin without relying on a central authority. One of the attractive features provided by blockchain technology is its append-only “immutability” feature, which means the stored data cannot be modified or manipulated by any means once it is validated in the blockchain ledger. Such immutability helps traceability, auditing, and non-repudiation, which builds decentralized trust among un-trusted parties. Despite that, immutability if misused could lead to the permanent existence of sensitive information and misinformation in the blockchain. Incidents like broadcasting illegal content have already taken their place in blockchain systems. Such incidents call for prompt solutions for mitigation. One emerging research theme, “redactable distributed ledgers” such as redactable blockchain provides approaches for modifying ledgers with certain controllability. This article aims to survey the current research landscape about redactable distributed ledgers. We will first describe the motivations behind redactable distributed ledgers. Compared to other relevant surveys, we comprehensively summarized and briefly explained the underlying technologies for supporting redactable distributed ledgers. We mainly focused on chameleon hash-based redactable blockchain structure and classifications, with detailed comparisons and illustrations. Further, we tackled new distributed ledger structures, including the new state-of-the-art block matrix structure. Furthermore, new applications that can be enabled by redactable distributed ledgers and future research directions are discussed in detail. This article emphasizes the motivation of utilizing the redactable distributed ledgers in several critical applications to mitigate misuse of immutability features threatening the original known design of distributed ledgers.