Polynomial chaos expansions have proven powerful for emulating responses of computational models with random input in a wide range of applications. However, they suffer from the curse of dimensionality, meaning the exponential growth of the number of unknown coefficients with the input dimension. By exploiting the tensor product form of the polynomial basis, low-rank approximations drastically reduce the number of unknown coefficients, thus providing a promising tool for effectively dealing with high-dimensional problems. In this paper, first, we investigate the construction of low-rank approximations with greedy approaches, where the coefficients along each dimension are sequentially updated and the rank of the decomposition is progressively increased. Furthermore, we demonstrate the efficiency of the approach in different applications, also in comparison with state-of-art methods of polynomial chaos expansions.