The Starshot Breakthrough Initiative established in 2016 sets an audacious goal of sending a spacecraft beyond our Solar System to a neighbouring star within the next half-century. Its vision for an ultralight spacecraft that can be accelerated by laser radiation pressure from an Earth-based source to similar to 20% of the speed of light demands the use of materials with extreme properties. Here we examine stringent criteria for the lightsail design and discuss fundamental materials challenges. We predict that major research advances in photonic design and materials science will enable us to define the pathways needed to realize laser-driven lightsails.