Diffusion-based communication refers to the transfer of information using molecules as message carriers whose propagation is governed by the laws of molecular diffusion. It has been identified that diffusion-based communication is one of the most promising solutions for end-to-end communication between nanoscale devices. In this paper, the design of a diffusion-based communication system considering stochastic signaling, arbitrary orders of channel memory, and noisy reception is proposed. The diffusion in the cases of one, two, and three dimensions are all considered. Three signal processing techniques for the molecular concentration with low computational complexity are proposed. For the detector design, both a low-complexity one-shot optimal detector for mutual information maximization and a near Maximum Likelihood (ML) sequence detector are proposed. To the best of our knowledge, our paper is the first that gives an analytical treatment of the signal processing, estimation, and detection problems for diffusion-based communication in the presence of ISI and reception noise. Numerical results indicate that the proposed signal processing technique followed by the one-shot detector achieves near-optimal throughput without the need of a priori information in both short-range and long-range diffusion-based communication scenarios, which suggests an ML sequence detector is not necessary. Furthermore, the proposed receiver design guarantees diffusion-based communication to operate without failure even in the case of infinite channel memory. A channel capacity of 1 bit per channel utilization can be ultimately achieved by extending the duration of the signaling interval.