Atomic force microscopy (AFM) uses a scanning process performed by a microcantilever probe to create a three dimensional image of a nano-scale physical surface. The dynamics of the AFM microcantilever motion and tip-sample force need to be understood to generate accurate images. Most AFMs use a laser system to take readings from the microcantilever. The bulky and expensive laser system can be replaced by piezoelectric actuators and sensors along with an electrical circuit. However, the dynamics of the piezoelectric microcantilever probe must be accurately modeled in order to generate accurate images and to take accurate readings when using the microcantilever for other applications. Additionally, minimizing the effect of nonlinearities in the dynamic response of the microcantilever allows for less calculation intensive software packages for AFMs without sacrificing accuracy. In this dissertation, the linear and nonlinear dynamics of a microcantilever probe in tapping mode AFM is investigated. First, different methods of including contact force in the linear equations of motion and boundary conditions are analyzed then compared to experimental results, which leads to the conclusion that including the contact force in the equation of motion and the inertial force due to the tip mass in the boundary conditions is the preferable method for most applications. The nonlinear vibrations of the tapping mode AFM microcantilever are investigated due to nonlinear contact force. The outcome shows that of the methods studied, the superior method of decreasing the nonlinearity effect is to find the optimal initial tip-sample distance and excitation force. Next, the effects of the nonlinear excitation force on the microcantilever's frequency and amplitude response are analytically studied. The results show a frequency shift in the response resulting from the force nonlinearities. The results of a sensitivity analysis show that parameters can be chosen such that the frequency shift is minimized. Additionally, a convergence analysis is used to determine the number of modes necessary to describe the motion of the microcantilever in tapping mode. It is determined that one mode is insufficient, and two modes are required and, for most applications sufficient.