Cultured meat (also referred to as cultivated meat or cell-based meat)—CM—is fabricated through the process of cellular agriculture (CA), which entails application of bioengineering, i.e., tissue engineering (TE) principles to the production of food. The main TE principles include usage of cells, grown in a controlled environment provided by bioreactors and cultivation media supplemented with growth factors and other needed nutrients and signaling molecules, and seeded onto the immobilization elements—microcarriers and scaffolds that provide the adhesion surfaces necessary for anchor-dependent cells and offer 3D organization for multiple cell types. Theoretically, many solutions from regenerative medicine and biomedical engineering can be applied in CM-TE, i.e., CA. However, in practice, there are a number of specificities regarding fabrication of a CM product that needs to fulfill not only the majority of functional criteria of muscle and fat TE, but also has to possess the sensory and nutritional qualities of a traditional food component, i.e., the meat it aims to replace. This is the reason that bioengineering aimed at CM production needs to be regarded as a specific scientific discipline of a multidisciplinary nature, integrating principles from biomedical engineering as well as from food manufacturing, design and development, i.e., food engineering. An important requirement is also the need to use as little as possible of animal-derived components in the whole CM bioprocess. In this review, we aim to present the current knowledge on different bioengineering aspects, pertinent to different current scientific disciplines but all relevant for CM engineering, relevant for muscle TE, including different cell sources, bioreactor types, media requirements, bioprocess monitoring and kinetics and their modifications for use in CA, all in view of their potential for efficient CM bioprocess scale-up. We believe such a review will offer a good overview of different bioengineering strategies for CM production and will be useful to a range of interested stakeholders, from students just entering the CA field to experienced researchers looking for the latest innovations in the field.