Global food systems are under significant pressure to provide enough food, particularly protein-rich foods whose demand is on the rise in times of crisis and inflation, as presently existing due to post-COVID-19 pandemic effects and ongoing conflict in Ukraine and resulting in looming food insecurity, according to FAO. Cultivated meat (CM) and cultivated seafood (CS) are protein-rich alternatives for traditional meat and fish that are obtained via cellular agriculture (CA) i.e., tissue engineering for food applications. Stem and progenitor cells are the building blocks and starting point for any CA bioprocess. This review presents CA-relevant vertebrate cell types and procedures needed for their myogenic and adipogenic differentiation since muscle and fat tissue are the primary target tissues for CM/CS production. The review also describes existing challenges, such as a need for immortalized cell lines, or physical and biochemical parameters needed for enhanced meat/fat culture efficiency and ways to address them.