Spikes are an important part of information transmission between neurons in the brain. There is biological evidence (such as rapid information processing in the electro sensory system of electric fish, the auditory system of echo-locating bats, and the visual system of flies) to prove the use of the precise timing of spikes for information coding, rather than only the firing rate. Recent research has shown the potential capability of spiking neural networks to model complex information processing in the brain. However, the exact learning mechanism in which the neuron is trained to fire at precise times remains an open problem. The aim of this project is to design novel biologically plausible learning algorithms for spiking neuron models at the neuron and the network levels. The first learning algorithm proposed is called BPSL (Biologically Plausible Supervised Learning algorithm), and it uses Spike Timing Dependent Plasticity (STOP) to adjust the weights. Additionally, the method sets strong teacher inputs to drive the neuron response and to prevent the silent neuron problem. Furthermore, local dendritic depolarization impacts the STOP in BPSL. A second approach called DL-ReSuMe, Delay Learning based Remote Supervised Method, is then proposed to merge weight adjustment and a delay shift approach to enhance learning performance and its biological plausibility. The weights are adjusted by LTP, Long-Term Potentiation, characteristic of BPSL and LTD, Long-Term Depression, of anti-STOP. The delay learning property helps DL-ReSuMe to solve the silent window problem. Then a third approach called EDL, Extended DL-ReSuMe, is proposed. Delays of appropriate groups of excitatory and inhibitory inputs are adjusted multiple times to find more accurate values for the delays. The delays and weights are adjusted cooperatively to construct a stable learning method. Subsequently, an approach termed Multi-DL-ReSuMe is proposed to extend DL-ReSuMe to a learning algorithm for a layer of spiking neurons for classification of spatiotemporal input patterns. The number of neurons in Multi-DL-ReSuMe is also increased to the number of the classes to improve its accuracy compared to a single neuron. Finally, a supervised learning algorithm for multilayer SNNs to train precisely the timing of multiple spikes is proposed. Weights of hidden and output neurons are adjusted in parallel to train the output neurons to fire at desired times. The delays of the output neurons are also adjusted in cooperation with the weight adjustments. EDL is used in the learning procedure of the multilayer network. Interactions between different layers of the network are governed through a biofeedback signal sent back by the output neurons. It uses another training procedure to prevent misclassification. During a misclassification, the learning algorithm adjusts the network parameters to train it to not fire close to the spikes that causes the misclassifications. The classification learning enables the proposed method to overcome the difficulty of learning of real world data. Simulation results are presented for all the proposed approaches to validate the propositions and benchmark against the current state of the art methods.