We study non-Markovian quantum systems for communication of quantum information. The quantum capacities of an exactly solvable spin-star system are analytically determined. Our results show that they strongly depend on the bath frequencies and system-bath coupling. The Ising spin-bath becomes noisy as the bath temperature is increased decreasing its capacity to transmit information. For random couplings and bath frequencies, recurrences of coherence are small in amplitude and rapidly die out. However, full periodic recurrences occur for equal coupling and frequencies. For small number of bath spins, the capacities are higher at low temperature. We also study two time-correlated non-unital channels for the transmission of classical and quantum information, provided limited or unlimited entanglement is shared prior to the communication. The noise over two successive uses of the channel is assumed to be time-correlated Markov noise. For amplitude-damping channel, the capacities exhibit strong dependence on channel memory. They decrease as the channel noise increases and reduce to zero when noise is maximum. However, our results show that the capacities are always non-zero, in the presence of channel memory. This is similar to the superactivation phenomenon. For perfect memory channel, the capacities acquire maximum value. Finally, we determine the entanglement-assisted classical capacity of generalized amplitude-damping channel. Our results show that depending on the channel noise and system-reservoir coupling, there exists a memory threshold. Below this threshold, the capacity decreases as channel memory increases. However, the capacity increases with memory beyond this threshold and is maximum for perfect memory channels.