A lightsail accelerated via directed energy is a candidate technology to send a probe into the deep space in a time period compatible with human life. The light emitted by a ground-based large-aperture phased laser array is directed onto the lightsail to produce a thrust by transferring the momentum of the incident photons. To achieve high efficiency propulsion, the lightsail must be characterized by extremely low mass, high reflectance and minimal absorption in the Doppler-shifted laser wavelength range, while high emissivity in the infrared is required for thermal management. Optimized multilayer structures allow ultralight spacecraft being accelerated by laser radiation pressure up to 20% of the light velocity, and eventually even above, as long as a compromise between efficiency and weight is achieved. Key aspect is the ability to survive to the temperature increase during the acceleration phase, which can be fateful for the system.