This paper investigates continuous-time (CT) signal acquisition as an activity-dependent and nonuniform sampling alternative to conventional fixed-rate digitisation. We demonstrate the applicability to biosignal representation by quantifying the achievable bandwidth saving by nonuniform quantisation to commonly recorded biological signal fragments allowing a compression ratio of 5 and 26 when applied to electrocardiogram and extracellular action potential signals, respectively. We describe several desirable properties of CT sampling, including bandwidth reduction, elimination/reduction of quantisation error, and describe its impact on aliasing. This is followed by demonstration of a resource-efficient hardware implementation. We propose a novel circuit topology for a charge-based CT analogue-to-digital converter that has been optimized for the acquisition of neural signals. This has been implemented in a commercially available 0.35 CMOS technology occupying a compact footprint of 0.12 mm2. Silicon verified measurements demonstrate an 8-bit resolution and a 4 kHz bandwidth with static power consumption of 3.75 W from a 1.5 V supply. The dynamic power dissipation is completely activity-dependent, requiring 1.39 pJ energy per conversion.