The current generation of optical surveys have opened up a new era of transient astronomy, but at the same time have introduced a new problem: our discovery capability has dramatically outpaced our follow-up capacity, such that less than 10 per cent of new transients receive a spectroscopic classification, let alone any scientific exploitation. The same problem is inherent in the gravitational wave follow-up programme, since the uncertainty in the sky position of any Advanced LIGO / Virgo detection is of the order of hundreds of degrees. Surveying this error box is not the biggest problem in identifying an electromagnetic counterpart: the real challenge is distinguishing the counterpart from the many unrelated transient events in the region. The rapid response and flexibility of robotic telescopes makes them ideal tools for these tasks, and accordingly the Liverpool Telescope is one of the world's leading facilities for transient follow-up. In this talk I will discuss the science programme and capabilities of the telescope, focusing on new capabilities such as the recently-deployed low-resolution spectrograph SPRAT, designed for the rapid classification of transient sources. The follow-up gap is going to increase in size by orders of magnitude as we move into the LSST era. I will therefore also provide an update on the development of Liverpool Telescope 2: a new 4-metre class telescope designed to meet the challenges of the coming decades of transient astronomy. Building on the strengths of the LT, LT2 will be a fully robotic telescope with a lightweight, fast-slewing design, providing a world-leading rapid response capability for efficient programmes of classification spectroscopy and the follow-up of fast-fading sources.