This talk presents a risk-averse topology optimization of linear elastic structures in the presence of uncertainty. We consider density-based formulations, in which the design is described by a continuous density field that takes values near one in solid regions and near zero in voids. The objective is to distribute a prescribed amount of material within a fixed design domain to minimize a performance metric of the design (e.g., compliance). The existence of solutions is ensured by regularization, which smooths the density field but introduces intermediate (“gray”) regions that require further processing. To obtain crisp, manufacturable designs, we often apply a Heaviside projection and incorporate erosion and dilation, yielding the classical robust formulation that optimizes against the worst-case realization of geometric perturbations. Even though effective for simple uncertainties, worst-case formulation becomes computationally impractical and often overly conservative when uncertainty stems from loading variability or rare failure scenarios such as support loss or localized damage. To address this, we replace the worst-case design with tail-sensitive risk measures, such as Conditional Value-at-Risk (CVaR) or Entropic Value-at-Risk (EVaR), which control the contribution of adverse events while preserving design freedom. For computationally efficient evaluation of CVaR and its sensitivities, we leverage its dual risk-envelope representation and couple it with importance sampling to concentrate simulations on the most influential tail realizations. The approach is demonstrated on 2D benchmarks and large-scale 3D problems.