Things can get subtle though - for example there are moduli functors that we care about which are not representable. One classical example is the moduli space of curves of genus $g$.
There are two solutions - 1) Weaken the definition of moduli space - this leads us to the definition of coarse moduli space $X$, which essentially means that whenever you have some family over $S$ you have a map $S\rightarrow X$, but there is no universal family over $X$ so given a map $S\rightarrow X$ you cannot pull this back to a family over $S$