Reconfigurable Intelligent Surfaces (RISs) are envisioned to play a key role in future wireless communications, enabling programmable radio propagation environments. They are usually considered as nearly passive planar structures that operate as adjustable reflectors, giving rise to a multitude of implementation challenges, including an inherent difficulty in estimating the underlying wireless channels. In this paper, we propose the concept of Hybrid RISs (HRISs), which do not solely reflect the impinging waveform in a controllable fashion, but are also capable of sensing and processing a portion of it via some active reception elements. We first present implementation details for this novel metasurface architecture and propose a simple model for its operation, when considered for wireless communications. As an indicative application of HRISs, we formulate and solve the individual channels identification problem for the uplink of multi-user HRIS-empowered systems. Our numerical results showcase that, in the high signal-to-noise regime, HRISs enable individual channel estimation with notably reduced amounts of pilots, compared to those needed when using a purely reflective RIS that can only estimate the cascaded channel.