Theorem. The category $\mathsf{Cat}$ of small categories is complete and cocomplete.
Proof. If $\{C_i\}_{i \in I}$ is a small diagram of categories, then define their limit $C$ by $\mathrm{Ob}(C) = \mathrm{lim}_i \mathrm{Ob}(C_i)$ and $\mathrm{Mor}(C) = \mathrm{lim}_i \mathrm{Mor}(C_i)$, with the obvious source, target and identity maps induced by the ones of the $C_i$ and the functoriality of $\lim$. Similarily, if $f = (f_i) : (x_i) \to (y_i)$ and $g = (g_i) : (y_i) \to (z_i)$ are composable morphisms, define $g \circ f = (g_i \circ f_i)$. It is easy to verify that $C$ is, in fact, a category, and that the obvious projections $C \to C_i$ satisfy the universal property of a limit.
The construction of $\mathrm{colim}_i C_i$ is more subtle. Consider the functor $\mathsf{Cat} \to \mathsf{Set}$, $C \mapsto \lim_i \hom(C_i,C)$. We want to show that it is representable, using Freyd's Representability Criterion (Mac Lane, Categories for the Working Mathematician, Theorem V.6.3). We already know that $\mathsf{Cat}$ is complete, and the functor is obviously continuous. Therefore, it suffices to verify the solution set condition.
Consider the cardinal $\kappa = \aleph_0 \cdot \sum\limits_{i \in I} \# \mathrm{Mor}(C_i)$.
Let $S$ be the set (!) of all categories whose object and morphism sets are subsets of $\kappa$. Observe that every category with $\# \mathrm{Mor} \leq \kappa$ is isomorphic to some category in $S$.
Let $\{F_i : C_i \to C\}$ be a compatible family of functors. Define a subcategory $C' \subseteq C$ as follows. Objects are those of the form $F_i(x)$ with $x$ is an object of $C_i$ and $i \in I$. A morphism in $C'$ is a morphism in $C$ which can be factored as $y_0 \to y_1 \to \dotsc \to y_n$, where each $y_j \to y_{j+1}$ lies in the image of some $F_i$. We allow $n=0$, which corresponds to the identity morphism. Clearly, $C'$ is a subcategory of $C$, and each $F_i$ factors through $C'$. The family $\{C_i \to C'\}$ is still compatible since $C'$ is a subcategory of $C$. Now basic cardinal arithmetic gives us $\# \mathrm{Mor}(C') \leq \sum_{n \in \mathbb{N}} \kappa^n = \kappa$. Hence, $C'$ is isomorphic to some object in $S$ and we are done. $ ~~\square$
Remark. The same proof can be used to show that $\mathrm{Mod}(T)$ is complete and cocomplete, where $T$ is an algebraic theory. But I don't think that $\mathsf{Cat}$ is algebraic, because the composition is only defined partially.
Perhaps $\mathrm{Mor} : \mathsf{Cat} \to \mathsf{Set}$ is monadic? A theorem of Linton (see Coequalizers in categories of algebras) says that $\mathsf{Mod}(T)$ is complete and cocomplete for every monad $T$ on $\mathsf{Set}$.
