Enriched locally presentable categories
The standard reference is Max Kelly’s 1982 paper “Structures defined by finite limits in the enriched setting”.
Perhaps the most unexpected thing is how well the theory works!
The theory is also developed further in two papers by Borceux, Quinteiro and Rosický: Enriched accessible categories MR1419612 and A theory of enriched sketches MR1624638. In the second one they say that they are particularly interested in case $\mathcal{V} = \text{Cat}$ and that they "intend to study this in a further publication", but I don't know of such a publication.
One remark about the standard theory of enriched accessible categories is that the enriching category $\mathcal{V}$ is typically assumed to itself be locally presentable. This ensures that for sufficiently large $\kappa$, there is a good notion of $\kappa$-small limits and $\kappa$-filtered colimits -- for instance, they should commute in $\mathcal{V}$ -- which makes for a good notion of "smallness" for $\mathcal{V}$-enriched categories.
In principle, though, one should be able to develop a theory of $(\mathcal{V},\Phi)$-accessible categories based on any good duality between a class $\Phi$ of $\mathcal{V}$-limits and a class $\Phi^\vee$ of $\mathcal{V}$-colimits that commute in $\mathcal{V}$. (By "good", I mean that a condition called soundness should be satisfied.) Conceptually, it's not necessary for $\mathcal{V}$ to be locally presentable or for $\Phi$ to be the class of $\kappa$-small $\mathcal{V}$-limits. For example, in the unenriched case, the duality between finite products and sifted colimits can be developed in perfect analogy to the theory of $\kappa$-accessible categories for fixed $\kappa$, and one recovers the usual theory of Lawvere theories. Some aspects of such a general theory of locally $(\mathcal{V},\Phi)$-presentable categories are touched on by Lack and Rosický.
What's missing from this story is an interesting notion of "raising the index of accessibility". In the standard enriched setup, where $\mathcal{V}$ is locally presentable, we have (just like the unenriched case) a hierarchy of $\Phi$'s to use -- $\kappa$-small $\mathcal{V}$-limits for arbitrarily large $\kappa$ -- and interesting theorems to state that rely on letting $\kappa$ vary. I simply don't know any good examples of such a hierarchy of $\Phi$'s besides the standard ones to motivate the development of the sort of general theory I'm driving at.