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We give explicit formulae for the continuous Hochschild and cyclic homologyand cohomology of certain topological algebras. To this end we show that, for acontinuous morphism $\phi: \X\to \Y$ of complexes of complete nuclear$DF$-spaces, the isomorphism of cohomology groups $H^n(\phi): H^n(\X) \toH^n(\Y)$ is automatically topological. The continuous cyclic-type homology andcohomology are described up to topological isomorphism for the followingclasses of biprojective $\hat{\otimes}$-algebras: the tensor algebra $E\hat{\otimes} F$ generated by the duality $(E, F, < \cdot, \cdot >)$ fornuclear Fr\'echet spaces $E$ and $F$ or for nuclear $DF$-spaces $E$ and $F$;nuclear biprojective K\"{o}the algebras $\lambda(P)$ which are Fr\'echet spacesor $DF$-spaces; the algebra of distributions $\mathcal{E}^*(G)$ on a compactLie group $G$.

Cyclic cohomology of certain nuclear Fr\'echet and DF algebras