Given a suitable class $\mathcal K$ of finite structures, a theorem of Fraïssé shows how to construct a special countable model, called the Fraïssé limit of $\mathcal K$: the unique dense $G_\delta$ (a.k.a., “generic”) isomorphism type in the space of all countable structures built from $\mathcal K$. Assuming the Continuum Hypothesis (CH), the same is true one cardinality higher: if $\mathcal K$ satisfies the hypotheses of Fraïssé’s Theorem, then there is a unique generic isomorphism type in the space of all size-$\mathfrak c$ structures built from $\mathcal K$, and furthermore, these special models of size $\mathfrak c$ have properties analogous to their corresponding Fraïssé limits. Some classes of finite structures do not satisfy the hypotheses of Fraïssé’s theorem, and these classes do not have Fraïssé limits. Interestingly, however, a class $\mathcal K$ may have no Fraïssé limit, but still have a unique size-$\mathfrak c$ generic model under CH. In other words, under CH, big generic models exist for any even broader range of classes than Fraïssé limits do, giving rise to Fraïssé-like uncountable structures with no true countable analogues. We will describe some aspects of the construction of these higher Fraïssé limits, and give several examples of familiar structures that can be understood in this way.