Let

$H$ and

$K$ be groups of order

$n$. A mapping

$f$ from

$H$ to

$K$ is called a planar function of degree

$n$ if for each

$h\in H-\left\{1\right\}$, the induced mapping

${f}_{h}:x\to f\left(hx\right)f{\left(x\right)}^{-1}$ is bijective. It is known that a planar function exists if and only if there exists an

$(n,n,n,1)$-relative difference set in

$H\times K$ relative to

$\left\{1\right\}\times K$. The author uses character theory to prove new results on the existence of planar functions from

${Z}_{n}$ to

${Z}_{n}$ and for the corresponding relative difference sets. In particular, the author shows that there are no planar functions from

${Z}_{pq}$ to

${Z}_{pq}$ where

$p$ and

$q$ are any primes and that except for 4 undecided cases, there is no planar function from

${Z}_{n}$ to

${Z}_{n}$ if

$n$ is not a prime and

$n\le 50,000$.