Universal metric spaces and extension dimension

For any countable CW-complex K and a cardinal number τ⩾ω we construct a completely metrizable space X(K,τ) of weight τ with the following properties: e-dimX(K,τ)⩽K, X(K,τ) is an absolute extensor for all normal spaces Y with e-dimY⩽K, and for any completely metrizable space Z of weight ⩽τ and e-dimZ⩽K the set of closed embeddings Z→X(K,τ) is dense in the space C(Z,X(K,τ)) of all continuous maps from Z into X(K,τ) endowed with the limitation topology. This result is applied to prove the existence of universal spaces for all metrizable spaces of given weight and with a given cohomological dimension.