Context. The Sardinia Radio Telescope (SRT) is the new 64 m dish operated by the Italian National Institute for Astrophysics (INAF).Its active surface, comprised of 1008 separate aluminium panels supported by electromechanical actuators, will allow us to observe at frequencies of up to 116 GHz. At the moment, three receivers, one per focal position, have been installed and tested: a 7-beam K-band receiver, a mono-feed C-band receiver, and a coaxial dual-feed L/P band receiver. The SRT was officially opened in September 2013, upon completion of its technical commissioning phase. In this paper, we provide an overview of the main science drivers for the SRT, describe the main outcomes from the scientific commissioning of the telescope, and discuss a set of observations demonstrating the scientific capabilities of the SRT. Aims. The scientific commissioning phase, carried out in the 2012-2015 period, proceeded in stages following the implementation and/or fine-tuning of advanced subsystems such as the active surface, the derotator, new releases of the acquisition software, etc. One of the main objectives of scientific commissioning was the identification of deficiencies in the instrumentation and/or in the telescope subsystems for further optimization. As a result, the overall telescope performance has been significantly improved. Methods. As part of the scientific commissioning activities, different observing modes were tested and validated, and the first astronomical observations were carried out to demonstrate the science capabilities of the SRT. In addition, we developed astronomeroriented software tools to support future observers on site. In the following, we refer to the overall scientific commissioning and software development activities as astronomical validation. Results. The astronomical validation activities were prioritized based on technical readiness and scientific impact. The highest priority was to make the SRT available for joint observations as part of European networks. As a result, the SRT started to participate (in shared-risk mode) in European VLBI Network (EVN) and Large European Array for Pulsars (LEAP) observing sessions in early 2014. The validation of single-dish operations for the suite of SRT first light receivers and backends continued in the following year, and was concluded with the first call for shared-risk early-science observations issued at the end of 2015. As discussed in the paper, the SRT capabilities were tested (and optimized when possible) for several different observing modes: imaging, spectroscopy, pulsar timing, and transients.