The flexoelectric (FxE) effect, where polarization is induced by a strain gradient, is universal in all insulators. As devices shrink to the micro and nano scale, large strain gradients can occur, and therefore the FxE effect can play a significant role in their electrical and mechanical properties. Also, the FxE effect can be exploited for novel device design paradigms such as piezoelectric "meta-materials'' constructed from nonpiezoelectric constituents, or mechanical switching of ferroelectric polarization. One of the crucial limitations to understanding and exploiting the FxE effect has been the lack of an efficient first-principles methodology to calculate all of the components of the bulk FxE tensor; the clamped-ion transverse and shear components in particular are problematic.  We have developed such a methodology based on density functional perturbation theory to calculate the full bulk, clamped-ion FxE tensor with unprecedented accuracy and efficiency. In this talk I will review the microscopic aspects of the FxE effect, describe our computational methodology, and provide results for some simple systems including cubic perovskite oxides.