Phytochemical analysis, the chemical investigation of plants, is a rapidly developing chemical discipline with various goals such as the determination of the substance classes and the isolation and qualitative and/or quantitative analyses of bioactive compounds or their structural analysis. The analysis of plants begins with optimizing extraction procedures because they play a key role in extraction outcomes, i.e., the isolation of bioactive compounds from natural products and the selection of the subsequent tests used in the analyses. The extraction is a critical step because it is necessary to extract the desired components of the complex natural matrix without destroying them. A wide range of techniques for the extraction of bioactive compounds from natural matrixes are currently available. This review aims to describe and compare the most commonly used methods, traditional/conventional and innovative/green. It will focus on the principles behind each method and on their strengths and limitations in order to help evaluate their suitability and economic feasibility, highlighting the advantages of new innovative techniques over conventional extraction procedures. Finally, the recent application of deep eutectic solvents as a new, eco-friendly method for the extraction of bioactive compounds from natural matrixes is also discussed.