MPEG-4 is an ISO/IEC standard, developed by the MPEG (Moving Picture Experts Group). The transmission of MPEG-4 compressed video over channels with different capacities may require a reduction in bit rate if the transmission media has a lower capacity than the capacity required by the bitstream. There are many different approaches to this problem of bit rate conversion. A feasible and effective method is transcoding. The emphasis in this thesis is on transcoding of MPEG-4 compressed video both in the pixel domain and in the DCT domain. In the pixel domain, (joint) transcoding of MPEG-4 compressed video with drift error correction is discussed in this thesis. In the DCT domain, a new DCT Coefficient Translation and Truncation Transformation Matrix (DCTTTM) based motion composition scheme is proposed, and several suboptimal approaches to reducing the computational complexity are discussed. In addition, frame-skipping transcoding of MPEG-4 compressed bitstreams in the DCT domain is addressed as well.