Nonnegative matrix factorization is an appealing technique for many audio applications. However, in it's basic form it does not use temporal structure, which is an important source of information in speech processing. In this paper, we propose NMF-based filtering and smoothing algorithms that are related to Kalman filtering and smoothing. While our prediction step is similar to that of Kalman filtering, we develop a multiplicative update step which is more convenient for nonnegative data analysis and in line with existing NMF literature. The proposed smoothing approach introduces an unavoidable processing delay, but the filtering algorithm does not and can be readily used for on-line applications. Our experiments using the proposed algorithms show a significant improvement over the baseline NMF approaches. In the case of speech denoising with factory noise at 0 dB input SNR, the smoothing algorithm outperforms NMF with 3.2 dB in SDR and around 0.5 MOS in PESQ, likewise source separation experiments result in improved performance due to taking advantage of the temporal regularities in speech.