Synthetic gene expression in engineered cells typically depends on the host cell’s limited pool of intracellular resources, which can lead to resource competition between native and engineered functions and may cause unanticipated effects in the host and synthetic gene circuit. For example, competition for shared resources may lead to cellular overload affecting physiological functions (that is, cellular burden) and negatively affect synthetic construct performance to the point of unreliability. This fundamental problem has implications for the use of synthetic genetic constructs in cell engineering for foundational research, therapy and bioprocessing. Resource competition has mainly been investigated in model bacteria systems, but also considerably affects eukaryotic systems, including mammalian cells. In this Review, we discuss resource competition beyond bacteria, outlining how it can lead to gene expression coupling, gene expression and metabolic burden. We also examine ways to quantify cellular burden in mammalian cells, and investigate circuit-centric and host-centric mitigation strategies, highlighting important implications of resource competition for cell engineering in therapeutic and bioproduction applications as well as in fundamental biology studies.