The action of the Virasoro algebra on integrable hierarchies of non-linear equations and on related objects (“Schrödinger” differential operators) is investigated. The method consists in pushing forward the Virasoro action to the wave function of a hierarchy, and then reconstructing its action on the dressing and Lax operators. This formulation allows one to observe a number of suggestive similarities between the structures involved in the description of the Virasoro algebra on the hierarchies and the structure of conformal field theory on the world-sheet. This includes, in particular, an “off-shell” hierarchy version of operator products and of the Cauchy kernel. In relation to matrix models, which have been observed to be effectively described by integrable hierarchies subjected to Virasoro constraints, I propose to define general Virasoro-constrained hierarchies also in terms of dressing operators, by certain equations which carry the information of the hierarchy and the Virasoro algebra simultaneously and which suggest an interpretation as operator versions of recursion/loop equations in topological theories. These same equations provide a relation with integrable hierarchies with quantized spectral parameter introduced recently. The formulation in terms of dressing operators allows a scaling (continuum limit) of discrete (i.e. lattice) hierarchies with the Virasoro constraints into “continuous” Virasoro-constrained hierarchies. In particular, the KP hierarchy subjected to the Virasoro constraints is recovered as a scaling limit of the Virasoro-constrained Toda hierarchy. The dressing operator method also makes it straightforward to identify the full symmetry algebra of Virasoro-constrained hierarchies, which is related to the family of W ∞( J) algebras introduced recently. For the KP hierarchy subjected to the Virasoro constraints depending on a parameter which may be interpreted as a conformal weight of an abstract bc system, we find the Borel subalgebra of W ∞( J), which we describe as an extension by the L 2 Virasoro generator of the “wedge”, or higher-spin, algebra B λ=J 2−J . Reductions to the generalized N-KdV hierarchies and the analysis of the corresponding constraints are also carried out. In particular, the W N algebra in the N-reduced case is related to W ∞ through a hamiltonian reduction and the sl( N) Kac-Moody algebra is observed.