Analog-to-digital converters (ADCs) allow physical signals to be processed using digital hardware. The power consumed in conversion grows with the sampling rate and quantization resolution, imposing a major challenge in power-limited systems. A common ADC architecture is based on sample-and-hold (S/H) circuits, where the analog signal is being tracked only for a fraction of the sampling period. In this paper, we propose the concept of eSampling ADCs, which harvest energy from the analog signal during the time periods where the signal is not being tracked. This harvested energy can be used to supplement the ADC itself, paving the way to the possibility of zero-power consumption and power-saving ADCs. We analyze the tradeoff between the ability to recover the sampled signal and the energy harvested, and provide guidelines for setting the sampling rate in the light of accuracy and energy constraints. Our analysis indicates that eSampling ADCs operating with up to 12 bits per sample can acquire bandlimited analog signals such that they can be perfectly recovered without requiring power from the external source. Furthermore, our theoretical results reveal that eSampling ADCs can in fact save power by harvesting more energy than they consume. To verify the feasibility of eSampling ADCs, we present a circuit-level design using standard complementary metal oxide semiconductor (CMOS) 65 nm technology. An eSampling 8-bit ADC which samples at 40 MHZ is designed on a Cadence Virtuoso platform. Our experimental study involving Nyquist rate sampling of bandlimited signals demonstrates that such ADCs are indeed capable of harvesting more energy than that spent during analog-to-digital conversion, without affecting the accuracy.