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I will present joint work with Brooke Shipley, in which we have defined a model category structure on the category of \( \Sigma^{\infty}X_+\)-comodule spectra such that the K-theory of the associated Waldhausen category of homotopically finite objects is naturally weakly equivalent to the usual Waldhausen K-theory of \( X\), \( A(X)\). I will describe the relation of this comodule approach to \( A(X)\) to the more familiar description in terms of \( \Sigma^\infty \Omega X_+\)-module spectra. I will also explain the construction and properties of the topological coHochschild homology of \( X\), which is a potentially interesting approximation to \( A(X)\).