Superdense coding and quantum teleportation are two phenomena which were not possible without prior entanglement. In superdense coding, one sends n bits of information using n/2 qubits in the presence of shared entanglement. However, we show that n bits of information cannot be sent with less than n bits of communication in LOCC protocols even in the presence of prior entanglement. This is an interesting result which will be used in the rest of this thesis. Quantum teleportation uses prior entanglement and classical communication to send an unknown quantum state. Remote state preparation (RSP) is the same distributed task, but in the case that the sender knows the description of the state to be sent, completely. We study the communication complexity of approximate remote state preparation in which the goal is to prepare an approximation of the desired quantum state. Jain showed that the worst-case error communication complexity of RSP can be bounded from above in terms of the maximum possible information in an encoding [18]. He also showed that this quantity is a lower bound for communication complexity of exact remote state preparation [18]. In this thesis, we characterize the worst-case error and average-case error communication complexity of remote state preparation in terms of non-asymptotic information-theoretic quantities. We also utilize the bound we derived for the communication complexity of LOCC protocols in the first part of the thesis, to show that the average-case error communication complexity of RSP can be much smaller than the worst-case.