Animal models have been the main resources for drug discovery and prediction of drugs’ pharmacokinetic responses in the body. However, noticeable drawbacks associated with animal models include high cost, low reproducibility, low physiological similarity to humans, and ethical problems. Engineered tissue models have recently emerged as an alternative or substitute for animal models in drug discovery and testing and disease modeling. In this review, we focus on skeletal muscle and cardiac muscle tissues by first describing their characterization and physiology. Major fabrication technologies (i.e., electrospinning, bioprinting, dielectrophoresis, textile technology, and microfluidics) to make functional muscle tissues are then described. Finally, currently used muscle tissue models in drug screening are reviewed and discussed