Hybrid automatic repeat request (HARQ) \cite{costello1983error} plays an important role in providing reliable and efficient data transmission. In wireless communications, the wireless channel may vary fast, due to the mobility of the transmitter/receiver and the channel. Forward error correction (FEC) and automatic repeat request (ARQ) are two basic techniques to control errors. FEC employs error correction coding, by adding parity bits to the information bits, to combat channel errors. ARQ allows the receiver to request a retransmission of the packet when an error is detected in the received packet. HARQ gives protection to the wireless transmission by combining FEC and ARQ. In typical HARQ systems, redundancy is added to the information bits, and a retransmission is performed until either the packet is successfully decoded, or a maximum number of transmissions is reached.The motivation to optimize the energy consumption of HARQ is the high energy consumption of wireless communications on mobile devices. Wireless devices usually have a limited battery life, and wireless communications consume the majority of the battery energy of mobile devices. One example is that 3G and Wifi units consume more than 50\% of the energy for some smart phones \cite{tawalbeh2016studying}. Another example is that battery depletion has been identified as one of the primary factors that limit the lifetime of wireless sensor networks \cite{verdone2010wireless}.Previous works on HARQ mainly use information-theoretic approach, which assumes that the number of bits in each transmission round is sufficiently large. This assumption does not necessarily hold for actual codes with finite length. Therefore, in this dissertation, we consider HARQ with actual codes. We use turbo-coded HARQ, since turbo codes are well-known capacity-approaching codes \cite{berrou1993near} and widely used in standards such as 3GPP Long-Term Evolution (LTE) \cite{3gpp2007mulltiplexing}. We study the energy optimization for HARQ in two scenarios: the energy optimization for incremental redundancy (IR) HARQ, and the energy optimization for HARQ in wireless video transmission. For IR HARQ, each retransmission contains additional parity bits beyond those of the previous transmissions. For the first scenario, we consider different cases of channel state information (CSI) at the transmitter: the transmitter has no knowledge of any CSI, or knows the CSI in previous transmission rounds through a perfect feedback channel, or knows both current and previous CSI. The transmitter decides the forward error correction code rate based on the CSI it has. We minimize the energy consumption of turbo-coded HARQ, subject to a packet loss rate constraint. Numerical results show that the energy consumption of HARQ decreases when more CSI information is available at the transmitter. We also compare IR combining with both Chase combining and the system without combining, and IR combining yields the least energy consumption.For the second scenario, we formulate the problem as maximizing the video quality, subject to a constraint on the wireless transmission energy consumption. We consider multiple parameters in multiple layers in a wireless video transmission system: transmit power, alphabet size, FEC code rate, maximum number of transmissions and unequal video data importance. An analytical framework is proposed to include these parameters, which allows us to divide this problem into two sub-problems: data transmission and unequal error protection (UEP) for video content. The problem is tackled by solving the two sub-problems, which are done by exhaustive search and convex optimization, respectively. Simulations of different videos show that the proposed scheme outperforms methods using conventional data transmission and/or unequal error protection. For example, in the low SNR region, there is a total gain of 4.8 to 5.6dB on the peak signal-to-noise ratio of the received video compared to video transmission using conventional HARQ without any video UEP
