This work presents some applications of stochastic spectral methods for structural topology optimization in the presence of uncertainties. This procedure, known as robust topology optimization, minimizes a combination of the mean and standard deviation of the objective function. For this purpose, a non-intrusive polynomial chaos expansion is integrated into a topology optimization algorithm to calculate the first two statistical moments of the mechanical model response. In order to address variabilities in the structural response, the uncertainties are considered in the loading and the material properties. In this proposed probabilistic formulation, uncertainties are represented as a set of random variables (e.g., magnitudes and directions of the loads) or as random fields (e.g., distributed loads and material properties). A non-Gaussian homogenous random field with a specified marginal distribution and covariance function is used to represent the material uncertainties because it ensures their physical admissibility. Nonlinearm memoryless transformation of a homogeneous Gaussian field is used for obtaining non-Gaussian fields. The Karhunen-Loève expansion is employed to provide a representation of the Gaussian field in terms of countable uncorrelated random variables. The sparse grid quadrature is considered for reducing the computational cost when computing the coefficients of the polynomial chaos expansion. Moreover, an efficient prediction (i.e., with a low computational cost) of the structural response under uncertainties is presented. Accuracy and applicability of the proposed methodology are demonstrated by means of several topology optimization examples of 2D continuum structures. The obtained results are in excellent agreement with the solutions obtained using the Monte Carlo method. Finally, conclusions are inferred and possible extensions of this work are proposed.