In a comment to François's answer, I point out that the least $\alpha$ such that $L_\alpha$ is a model of $\mathsf{ZFC}$ is countable. In what follows, "model" means "transitive model of $\mathsf{ZFC}$."

If $M$ is transitive, then $L^M=L_\beta\models\mathsf{ZFC}$ for $\beta=\mathsf{ORD}\cap M$, so the least height of a model is countable. Moreover, any model $M$ proves that there is a bijection between each level of its cumulative hierarchy, and one of its ordinals, so if $M$ has height $\kappa$, so is $\kappa$ its size. This proves that the existence of a model does not imply the existence of an uncountable one: If $M$ has least height, let $\alpha_0$ be least such that $L_{\alpha_0}$ is a model, and $\alpha_0$ is larger than the height of $M$ ($\alpha_0$ could be $\mathsf{ORD}$). We see that in $L_{\alpha_0}$ the height of $M$ is countable and there are no models of height larger than the height of $M$.

Asaf asked whether this generalizes, that is, whether for each $\kappa$ the existence of models of size $\kappa$ does not imply the existence of models of larger size. That this is indeed the case follows from extending the argument from the previous paragraph: Let $L_\alpha$ be a model of height (and size) $\kappa$, and let $\alpha_0$ be such that $\alpha<\alpha_0$ and $L_{\alpha_0}$ is a model. We may as well assume that $\alpha_0$ exists (that is, it is an ordinal), or else there is nothing to prove. Now let $X$ be an elementary substructure of $L_{\alpha_0}$ containing both $L_\alpha\cup\{L_\alpha\}$ (as a subset) and a bijection between $\alpha$ and $|\alpha|^{L_{\alpha_0}}$, and of size $\kappa$, which exists by a standard application of the Lowenheim-Skolem argument. The transitive collapse of $X$ is $L_\beta$ for some $\beta$, and has size $\kappa$. This means that if $\alpha_0$ is least (so the collapse of $X$ is again $L_{\alpha_0}$), then $L_{\alpha_0}$ is a model of $\mathsf{ZFC}$ plus the assertion that there are no set models of height (and therefore size) larger than $|\alpha|=\kappa$.

Without choice, I do not know whether models of $\mathsf{ZF}$ of height $\kappa$ must have size $\kappa$.

[**Edit:** In response to the last paragraph above, Joel and Asaf pointed out some results. I am including them here, to increase visibility.]

In

Ali Enayat. *Models of set theory with definable ordinals*, Arch. Math. Logic **44 (3)**, (2005), 363–385. MR2140616 (2005m:03098),

Ali discusses some results about transitive models of $\mathsf{ZF}$ that show that the situation is much more subtle than in the presence of choice. One of the most incredible results is due to Friedman, in

Harvey Friedman. *Large models of countable height*, Trans. Amer. Math. Soc. **201** (1975), 227–239. MR0416903 (54 #4966).

Let me quote from Harvey's paper:

The first examples of transitive models of $\mathsf{ZF}$ of power $\omega_1$ with countably many ordinals were constructed by Cohen. Later Easton, Solovay, and Sacks showed that every countable transitive model of $\mathsf{ZF}$ has an ordinal-preserving extension satisfying $\mathsf{ZF}$, of power $2^{\omega}$. We prove here that every countable transitive model $M$ of $\mathsf{ZF}$ has an ordinal preserving extension satisfying $\mathsf{ZF}$, of power $\beth_{M\cap\mathsf{ORD}}$.

Harvey's argument uses forcing. For his first result, given $M$ a countable transitive model of $\mathsf{ZF}$, he says that $x\subset\omega^\omega$ is $M$-generic iff any finite sequence of distinct elements of $x$ is $M$-generic (for the product of the appropriate number of copies of Cohen forcing), $x$ is infinite, and dense. The models he builds are of the form $M(x)$ so, in particular, they are transitive. He shows that there are $M$-generics $x$ of size continuum with $M(x)$ a model of $\mathsf{ZF}$ (and $M(x)$ has the same height as $M$). He then builds on the machinery introduced here, and proves that, starting with an $M$-generic $x$, a family of sets $C_\alpha$, $\alpha\lt M\cap\mathsf{ORD}$, can be found with $|C_\alpha|\ge\beth_\alpha$, and such that $M[(C_\alpha)_\alpha]$, properly defined, is a model of $\mathsf{ZF}$ of the claimed size.

Ali builds on this results to produce *Paris models* of $\mathsf{ZF}$, that is, models $M$ all of whose ordinals are first order definable in $M$. In prior work, he had shown that from the assumption that $L$ satisfies that there are uncountable transitive models of $\mathsf{ZFC}$, it follows that there are unboundedly many $\alpha<\omega_1^L$ such that $L_\alpha$ is Paris. He shows now that from the same assumption, we have that for every infinite $\kappa$ there are Paris models of $\mathsf{ZF}$ of size $\kappa$; this uses Harvey's result, since generic (or simply, ordinal preserving) extensions of $L_\alpha$ are Paris if $L_\alpha$ itself is Paris. It follows that there is a complete extension of $\mathsf{ZF}$ admiting in $L$ Paris models of size $\beth_\alpha$ for each countable $\alpha$. The theory, including the requirement that its models are Paris, can be described in $L_{\omega_1\omega}$. Since the Hanf number of this logic is $\beth_{\omega_1}$, the result follows.

This produces large transitive models indeed, in view of a result of Paris: If a completion $T$ of $\mathsf{ZF}$ has a well-founded model, then *every* Paris model of $T$ is well-founded.

As pointed out by Mohammad here, Solovay's construction referenced in Friedman's answer (and a clear influence in Friedman's argument) can be found in

Ulrich Felgner. *Choice functions on sets and classes*. In **Sets and classes (on the work by Paul Bernays)**, pp. 217–255. Studies in Logic and the Foundations of Math., Vol. 84, North-Holland, Amsterdam, 1976. MR0424566 (54 #12525).