In this paper, we report the first LED-pumped transition-metal-doped laser oscillator and amplifier based on an alexandrite crystal (Cr<sup>3+</sup>:BeAl<sub>2</sub>O<sub>4</sub>). A Ce:YAG luminescent concentrator illuminated by blue LEDs is used to reach higher pump powers than with LEDs alone. The luminescent 200-mm-long-composit luminescent concentrator involving 2240 LEDs can delivers up to 268 mJ for a peak irradiance of 8.5 kW/cm<sup>2</sup>. In oscillator configuration, an LED-pumped alexandrite laser delivering an energy of 2.9 mJ at 748 nm in free running operation is demonstrated. In the cavity, we measured a double pass small signal gain of 1.28, in good agreement with numerical simulations. As amplifier, the system demonstrated to boost a CW Ti:sapphire laser by a factor of 4 at 750 nm in 8 passes with a large tuning range from 710 nm to 800 nm.