Cell-to-cell variability originating, for example, from the intrinsic stochasticity of gene expression, presents challenges for designing synthetic gene circuits that perform robustly. Conversely, synthetic biology approaches are instrumental in uncovering mechanisms underlying variability in natural systems. With a focus on reducing noise in individual genes, the field has established a broad synthetic toolset. This includes noise control by engineering of transcription and translation mechanisms either individually, or in combination to achieve independent regulation of mean expression and its variability. Synthetic feedback circuits use these components to establish more robust operation in closed-loop, either by drawing on, but also by extending traditional engineering concepts. In this perspective, we argue that major conceptual advances will require new theory of control adapted to biology, extensions from single genes to networks, more systematic considerations of origins of variability other than intrinsic noise, and an exploration of how noise shaping, instead of noise reduction, could establish new synthetic functions or help understanding natural functions.