Living systems are inherently stochastic and operate in a noisy environment, yet despite all these uncertainties, they perform their functions in a surprisingly reliable way. The biochemical mechanisms used by natural systems to tolerate and control noise are still not fully understood, and this issue also limits our capacity to engineer reliable, quantitative synthetic biological circuits. We study how representative models of biochemical systems propagate and attenuate noise, accounting for intrinsic as well as extrinsic noise. We investigate three molecular noise-filtering mechanisms, study their noise-reduction capabilities and limitations, and show that nonlinear dynamics such as complex formation are necessary for efficient noise reduction. We further suggest that the derived molecular filters are widespread in gene expression and regulation and, particularly, that microRNAs can serve as such noise filters. To our knowledge, our results provide new insight into how biochemical networks control noise and could be useful to build robust synthetic circuits.