The ability to engineer synthetic circuits and devices in mammalian cells has enabled a multitude of exciting applications in industrial biotechnology and medical therapy. In spite of the recent availability of powerful genome engineering tools such as CRISPR-Cas9, the process of designing and implementing functioning genetic circuits remains painstakingly slow and fraught with inexplicable failures. The unexpected divergence between intended and actual function of synthetic circuits can be attributed to several factors, most notably the contextual background in which these circuits operate. In particular, t he dependence of synthetic circuits on cellular resources which are limited leads to unintended dynamic couplings between the various exogenous components of the circuit, as well as with endogenous components of the host cell. The consumption of these resources by synthetic circuits thus exerts a burden on the host cell that reduces its capacity to support additional circuits, potentially resulting in counter-intuitive functional behaviors and detrimental effects on host physiology. In spite of its critical importance, gene expression burden in mammalian cells remains largely unstudied. Here, we comprehensively investigate the impact of host resource limitations on synthetic constructs in mammalian cells. We show that effects of both transcriptional and translational resource limitations can be observed and that each can lead to the coupling of independent, co-expressed synthetic genes, which in turn imposes trade-offs in their expression and diminishes performance. We next explore the role of post-transcriptional regulators, such as microRNAs (miRNAs) and RNA binding proteins (RBPs) and show that they can redistribute resources in a way that limits burden-induced coupling effects. To quantify and predict the influence of burden on gene expression in engineered cells, we describe a modelling framework that allows to incorporate the effect of limited resources into classical models of gene expression. Based on this framework, we identify network topologies that mitigate burden and then implement these topologies using endogenous and synthetic miRNA-based circuits that buffer the expression of genes of interest from fluctuations in cellular resources. Among other regulators, microRNAs can conveniently tailor synthetic device regulation in different cell lines and tissues, as well as during dynamic changes of cellular states and downstream information processing, or in pathological conditions. This study thus establishes a foundation for context-aware predictions of in vivo synthetic circuit performance and paves the way towards a more rational synthetic construct design in mammalian cells.