This paper explores the relationship between mixed-state entanglement and quantum error correction. It proves that entanglement purification protocols (EPP) and quantum error-correcting codes (QECC) are closely related. Specifically, an EPP involving one-way classical communication on a mixed state M yields a QECC with rate Q = D, and vice versa. The paper compares the amount of entanglement required to prepare a mixed state M with the amounts that can be distilled from it using EPPs with one- and two-way classical communication. It provides an exact expression for the entanglement of a Bell-diagonal state. The paper also discusses the role of entanglement in quantum information theory, showing how it is used to transmit quantum states through noisy channels. It introduces three entanglement measures for mixed states: D1(M), D2(M), and E(M), with E(M) being the entanglement of formation. The paper shows that E(M) is the minimum expected entanglement of any ensemble of pure states realizing M. It also proves that E(M) is nonincreasing under local operations and classical communication. The paper presents a family of one-way entanglement purification protocols and corresponding quantum error-correcting codes, as well as two-way protocols that can be used to transmit quantum states through noisy channels. It shows that certain noisy quantum channels, such as a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way communication is available. The paper also exhibits a family of codes based on universal hashing that can achieve an asymptotic Q (or D) of 1 - S for simple noise models, where S is the error entropy. It also presents a specific, simple 5-bit single-error-correcting quantum block code. The paper concludes by discussing the relationship between mixed states and quantum channels, and the implications for quantum error correction. It also reviews several important remaining open questions in the field.This paper explores the relationship between mixed-state entanglement and quantum error correction. It proves that entanglement purification protocols (EPP) and quantum error-correcting codes (QECC) are closely related. Specifically, an EPP involving one-way classical communication on a mixed state M yields a QECC with rate Q = D, and vice versa. The paper compares the amount of entanglement required to prepare a mixed state M with the amounts that can be distilled from it using EPPs with one- and two-way classical communication. It provides an exact expression for the entanglement of a Bell-diagonal state. The paper also discusses the role of entanglement in quantum information theory, showing how it is used to transmit quantum states through noisy channels. It introduces three entanglement measures for mixed states: D1(M), D2(M), and E(M), with E(M) being the entanglement of formation. The paper shows that E(M) is the minimum expected entanglement of any ensemble of pure states realizing M. It also proves that E(M) is nonincreasing under local operations and classical communication. The paper presents a family of one-way entanglement purification protocols and corresponding quantum error-correcting codes, as well as two-way protocols that can be used to transmit quantum states through noisy channels. It shows that certain noisy quantum channels, such as a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way communication is available. The paper also exhibits a family of codes based on universal hashing that can achieve an asymptotic Q (or D) of 1 - S for simple noise models, where S is the error entropy. It also presents a specific, simple 5-bit single-error-correcting quantum block code. The paper concludes by discussing the relationship between mixed states and quantum channels, and the implications for quantum error correction. It also reviews several important remaining open questions in the field.