This paper introduces the concept of signals with a finite rate of innovation (ρ), which refers to the number of degrees of freedom per unit of time. Examples include streams of Diracs, nonuniform splines, and piecewise polynomials. Unlike traditional bandlimited signals, these signals are not bandlimited but can still be sampled uniformly at or above their rate of innovation using appropriate kernels, allowing for perfect reconstruction. The paper presents sampling theorems for various signal classes, including periodic and finite-length streams of Diracs, nonuniform splines, and piecewise polynomials, using kernels such as sinc, Gaussian, and spline. The key technique involves identifying the innovative part of a signal using annihilating filters, which are well-known in spectral analysis and error-correction coding. The paper also discusses applications in signal processing, communications, and biological systems. The results show that signals with finite rate of innovation can be sampled and reconstructed using appropriate kernels and algorithms, even when they are not bandlimited. The paper concludes with a discussion of the implications of these results for signal processing and communications systems.This paper introduces the concept of signals with a finite rate of innovation (ρ), which refers to the number of degrees of freedom per unit of time. Examples include streams of Diracs, nonuniform splines, and piecewise polynomials. Unlike traditional bandlimited signals, these signals are not bandlimited but can still be sampled uniformly at or above their rate of innovation using appropriate kernels, allowing for perfect reconstruction. The paper presents sampling theorems for various signal classes, including periodic and finite-length streams of Diracs, nonuniform splines, and piecewise polynomials, using kernels such as sinc, Gaussian, and spline. The key technique involves identifying the innovative part of a signal using annihilating filters, which are well-known in spectral analysis and error-correction coding. The paper also discusses applications in signal processing, communications, and biological systems. The results show that signals with finite rate of innovation can be sampled and reconstructed using appropriate kernels and algorithms, even when they are not bandlimited. The paper concludes with a discussion of the implications of these results for signal processing and communications systems.