Sampling theory has experienced a strong research revival over the past decade, which led to a generalization of Shannon's original theory and development of more advanced formulations with immediate relevance to signal processing and communications. For example, it was recently shown that it is possible to develop exact sampling schemes for a large class of non-bandlimited signals, namely, certain signals with finite rate of innovation. A common feature of such signals is that they have a parametric representation with a finite number of degrees of freedom and can be perfectly reconstructed from a finite number of samples. The goal of this thesis is to advance the sampling theory for signals of finite rate of innovation and consider its possible extensions and applications. In the first part of the thesis, we revisit the sampling problem for certain classes of such signals, including non-uniform splines and piecewise polynomials, and develop improved schemes that allow for stable and precise reconstruction in the presence of noise. Specifically, we develop a subspace approach to signal reconstruction, which converts a nonlinear estimation problem into the simpler problem of estimating the parameters of a linear model. This provides an elegant and robust framework for solving a large class of sampling problems, while offering more flexibility than the traditional scheme for bandlimited signals. In the second part of the thesis, we focus on applications of our results to certain classes of nonlinear estimation problems encountered in wideband communication systems, most notably ultra-wideband (UWB) systems, where the bandwidth used for transmission is much larger than the bandwidth or rate of information being sent. We develop several frequency domain methods for channel estimation and synchronization in UWB systems, which yield high-resolution estimates of all relevant channel parameters by sampling a received signal below the traditional Nyquist rate. We also propose algorithms that are suitable for identification of more realistic UWB channel models, where a received signal is made up of pulses with different pulse shapes. Finally, we extend our results to multidimensional signals, and develop exact sampling schemes for certain classes of parametric non-bandlimited 2-D signals, such as sets of 2-D Diracs, polygons or signals with polynomial boundaries.