In a similar vein to Todd Trimble's remark on elementary toposes, let's recall the following fact:
Theorem. An accessible category is complete if and only if it is cocomplete.
The quickest proof I know of uses the accessible adjoint functor theorem. The curious thing is this: from a theoretical perspective, the cocompleteness hypothesis is more useful; yet in practice it is the completeness hypothesis that is easiest to verify. For instance, it is easy to see that many categories of essentially algebraic structures (e.g. groups, rings, categories) are accessible and complete, but the construction of colimits in these categories is often non-obvious.
The only instance of the "if" direction I am aware of is a corollary of Giraud's theorem: any category satisfying Giraud's axioms is necessarily complete. This is obvious if we assume Giraud's theorem, because any such category is a topos of sheaves on a Grothendieck site, hence, a reflective subcategory of a complete category.