This paper proposes two types of base isolation systems with tuned inerter dampers (TID) composed of a spring, an inerter and a dashpot in serial or parallel to enhance the seismic isolation performance of the traditional base-isolation system. The TID isolation systems are optimally tuned using the H 2 norm criterion to achieve the best RMS vibration performance under white-noise random excitation. The TID frequency ratio and damping ratio are defined as the design parameters, whose optimal values are analytically derived for the undamped primary system and numerically verified. The results show that the optimum exists in the serial TID isolation system (inerter and dashpot in serial layout), while in the parallel TID isolation system (inerter and dashpot in parallel layout) larger TID stiffness and larger TID damping are preferred in practice to achieve better isolation performance. The isolation performance of the H 2 optimal TID systems is also compared with that of the H ∞ optimized systems obtained numerically. The results suggest that the optimal serial TID system has overall better isolation performance over the broad frequency range, which is preferable for seismic vibration isolation because seismic waves usually contain a wide range of frequency components. The parallel TID system cannot be tuned optimally for practical structures, nevertheless, it still achieves better isolation performance at the resonant vibrations by an appropriate selection of the design parameters. The influence of the structural parameters on the tuning parameters are studied. Numerical case studies are conducted in comparison with the traditional and tuned mass damper (TMD) isolation systems for a five-story laboratory building prototype. The potential power in the TID isolations that could be converted to usable electricity in earthquakes are also examined.