Speech quality and intelligibility might significantly deteriorate in the presence of background noise, especially when the speech signal is subject to subsequent processing. In particular, speech coders and automatic speech recognition (ASR) systems that were designed or trained to act on clean speech signals might be rendered useless in the presence of background noise. Speech enhancement algorithms have therefore attracted a great deal of interest in the past two decades. In this paper, we present a class of Kaiman filter-based algorithms with some extensions, modifications, and improvements of previous work. The first algorithm employs the estimate-maximize (EM) method to iteratively estimate the spectral parameters of the speech and noise parameters. The enhanced speech signal is obtained as a byproduct of the parameter estimation algorithm. The second algorithm is a sequential, computationally efficient, gradient descent algorithm. We discuss various topics concerning the practical implementation of these algorithms. Extensive experimental study using real speech and noise signals is provided to compare these algorithms with alternative speech enhancement algorithms, and to compare the performance of the iterative and sequential algorithms.