Decoding DNA symbols using next-generation sequencers was a major breakthrough in genomic research. Despite the many advantages of next-generation sequencers, e.g., the high-throughput sequencing rate and relatively low cost of sequencing, the assembly of the reads produced by these sequencers still remains a major challenge. In this review, we address the basic framework of next-generation genome sequence assemblers, which comprises four basic stages: preprocessing filtering, a graph construction process, a graph simplification process, and postprocessing filtering. Here we discuss them as a framework of four stages for data analysis and processing and survey variety of techniques, algorithms, and software tools used during each stage. We also discuss the challenges that face current assemblers in the next-generation environment to determine the current state-of-the-art. We recommend a layered architecture approach for constructing a general assembler that can handle the sequences generated by different sequencing platforms.