Recovering sparse signals from linear measurements has demonstrated outstanding utility in a vast variety of real-world applications. Compressive sensing is the topic that studies the associated raised questions for the possibility of a successful recovery. This topic is well-nourished and numerous results are available in the literature. However, their dispersity makes it challenging and time-consuming for readers and practitioners to quickly grasp its main ideas and classical algorithms, and further touch upon the recent advancements in this surging field. Besides, the sparsity notion has already demonstrated its effectiveness in many contemporary fields. Thus, these results are useful and inspiring for further investigation of related questions in these emerging fields from new perspectives. In this survey, we gather and overview vital classical tools and algorithms in compressive sensing and describe significant recent advancements. We conclude this survey by a numerical comparison of the performance of described approaches on an interesting application.