Fast Radio Bursts with Apertif

This thesis describes searches for, and studies of, pulsars (PSRs) and fast radio bursts (FRBs). We developed novel instrumentation and used it to find and investigate new transients, in real time.
We present a search for pulsars in subdwarf-B binary systems. None were detected. However, we did discover a new, unassociated, giant-pulse emitting pulsar, PSR J0533–4524. We explore possible reasons for our non-detections, and discuss the characteristics of the discovered source.
We discovered an FRB with a low dispersion measure (DM). That is important because the source must be nearby. Similar low-DM, ultra-bright FRBs may be detected in telescope sidelobes in the future, making careful modelling of multi-beam instrument beam patterns of utmost importance.
We outline the design and commissioning of the Apertif Radio Transient System (ARTS), the new time-domain backend of the Westerbork Synthesis Radio Telescope (WSRT). With ARTS, the WSRT can perform real-time searches for radio transients over the full 8 square degree field-of-view provided by Apertif.
We present an ARTS observing campaign on the first two discovered repeating FRBs, R1 and R2. The detection of 30 bursts from R1, and non-detection of R2, provide improved constraints on the repeat rate and clustered nature of their bursts.
Lastly, we present the discovery of nine new FRBs with ARTS. We discuss the burst properties and localisation, and prospects for counterpart identification. The ARTS detection rate of one burst every 5 days is currently the highest in the world at 1.4 GHz.