We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of biflat topological algebras. We show that, for a continuous morphism $\varphi:{\mathcal X}^*\rightarrow{\mathcal Y}^*$ of complexes of complete nuclear DF-spaces, the isomorphism of cohomology groups $H^n(\varphi):H^n({\mathcal X}^*)\rightarrow H^n({\mathcal Y}^*)$ is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective topological algebras: the algebra of smooth functions $\mathcal{E}(G)$ on a compact Lie group G, the algebra of distributions $\mathcal{E}^*(G)$ on a compact Lie group G; the tensor algebra $E\hat{\otimes} F$ generated by the duality  $(E, F, <\cdot, \cdot>)$ for nuclear 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 spaces or DF-spaces.