Some critical components, such as bearings, gears, impellers, etc., are widely used in various machines. Their faults accelerate failures of other components and finally result in machine breakdowns. To prevent any unexpected machine breakdowns and accidents, early faults of these critical components should be detected as soon as possible. After that, once early faults are found, performance degradation assessment and remaining useful life estimation should be conducted to maximize lifetime of these critical components. In the past decades, vibration analysis has proven to be one of the most effective and efficient methods for fault diagnosis and prognosis of critical components because vibration signals can be conveniently collected from transducers installed on the casing of machines. Due to interruptions caused by heavy noises and other vibration components, vibration analysis is still extremely challenging. In this thesis, some advanced signal processing methods are respectively proposed for fault diagnosis and prognosis of critical components including bearings and impellers. First, bearings are one of the most common components used in machines to support rotation shafts. According to statistics, bearings are prone to suffer faults. For bearing fault diagnosis, it is necessary to separate bearing fault signals from other strong low-frequency periodic signals, such as periodic fault signals caused by imbalance, misalignment and eccentricity. This is the topic related to the concept of blind fault component separation, which aims to decompose vibration signals to periodic signals and random signals. To achieve blind fault component separation, two algorithms including a new blind fault separation algorithm and an enhanced empirical mode decomposition method are proposed. The new blind fault separation algorithm aims to use kurtosis and a smoothness index to extract periodic fault signals and bearing fault signals, respectively. The enhanced empirical mode decomposition method does not need those two objective functions to guide signal decomposition. The enhanced empirical mode decomposition method adaptively decomposes vibration signals to periodic signals and random signals. Second, even though bearing fault signals are separated from other strong lowfrequency periodic signals, heavy noises still exist in bearing fault signals. To enhance the signal to noise ratios of bearing fault signals and facilitate identification of different bearing faults, three algorithms including an enhanced kurtogram, a morphogram and an adaptive wavelet stripping algorithm are proposed. The enhanced kurtogram aims to locate and retain one of bearing resonant frequency bands for bearing fault diagnosis. The morphogram is a new concept that aims to find the optimal flat length used in morphological analysis and it can be used to extract envelope signals and retain cyclic bearing fault features. Besides, the morphogram can make bearing fault signals sparse, which means that only a few of coefficients are used to represent bearing fault signals. The adaptive wavelet stripping algorithm can well extract simulated transients from bearing fault signals and reconstruct these simulated transients to reflect the random characteristic of bearing fault signals. To fast diagnose early bearing faults, it is recommended to use the enhanced kurtogram because only a few wavelet packet filters are involved in this algorithm and the calculation time of the enhanced kurtogram is short. To achieve better visual inspection performance, the morphogram and the adaptive wavelet stripping algorithm can be conducted. However, it should be noted that the calculation times of the two methods are extensive because they aim to find the optimal parameters used in the two methods. In other words, if on-line fault diagnosis is required, the enhanced kurtogram is recommended to be conducted. If off-line fault diagnosis is required, the morphogram and the adaptive wavelet stripping algorithm can be used to analyze bearing fault signals. Thirdly, once early faults of key components are detected, performance degradation assessment and estimation of remaining useful life of these key components should be conducted. In this thesis, besides early fault diagnosis of bearings, performance degradation assessment and remaining useful life estimation of slurry pump impellers are investigated. Slurry pump impellers are one of the most common components used in slurry pumps to pump mixtures of abrasive and erosive liquids and solids, which cause constant wear of slurry pump impellers and result in slurry pump breakdowns. To monitor and evaluate the health condition of slurry pump impeller, a prognostic method for impeller performance degradation assessment and remaining useful life estimation is proposed. Impeller performance degradation assessment aims to track the current health condition of slurry pump impellers. Based on the current health condition, impeller remaining useful life could be estimated to make preventive maintenance and reduce impeller replacement cost. The proposed prognostic method is based on a general sequential Monte Carlo method, especially a general particle filter, which uses an amount of random particles with their associated weights to approximate posterior probability density functions, because the general particle filter can be applied to solve non-linear and non-Gaussian state space models. An estimate of remaining useful life of slurry pump impeller can be inferred from the posterior probability density functions. The uncertainty of the estimate can be established by calculating its the confidence intervals. Keywords: signal processing; bearing; impeller; fault diagnosis; prognosis; vibration analysis; remaining useful life; condition monitoring.