In recent work, Hess and Shipley have defined topological coHochschild homology (coTHH) for coalgebra spectra. In this talk, I will discuss coTHH for coalgebra spectra and dg-coalgebras, and give some computational results in about cofree coalgebras in both contexts. In the spectral world, these follow from an analogue of the Bokstedt spectral sequence for computing topological Hochschild homology, which I will also discuss. This is joint work with Gerhardt, Hogenhaven, Shipley, and Ziegenhagen.