  Koll\'ar introduced in [Ko08] the surfaces
$$(x_1^{a_1}x_2+x_2^{a_2}x_3+x_3^{a_3}x_4+x_4^{a_4}x_1=0)\subset
\mathbb{P}(w_1,w_2,w_3,w_4)$$ where $w_i=W_i/w^*$,
$W_i=a_{i+1}a_{i+2}a_{i+3}-a_{i+2}a_{i+3}+a_{i+3}-1$, and
$w^*=$gcd$(W_1,\ldots,W_4)$. The aim was to give many interesting examples of
$\mathbb{Q}$-homology projective planes. They occur when $w^*=1$. For that
case, we prove that Koll\'ar surfaces are Hwang-Keum [HK12] surfaces. For
$w^*>1$, we construct a geometrically explicit birational map between Koll\'ar
surfaces and cyclic covers $z^{w^*}=l_1^{a_2 a_3 a_4} l_2^{-a_3 a_4} l_3^{a_4}
l_4^{-1}$, where $\{l_1,l_2,l_3,l_4\}$ are four general lines in
$\mathbb{P}^2$. In addition, by using various properties on classical Dedekind
sums, we prove that:
  (a) For any $w^*>1$, we have $p_g=0$ iff the Koll\'ar surface is rational.
This happens when $a_{i+1} \equiv 1$ or $a_{i}a_{i+1} \equiv -1 ($mod $w^*)$
for some $i$.
  (b) For any $w^*>1$, we have $p_g=1$ iff the Koll\'ar surface is birational
to a K3 surface. We classify this situation.
  (c) For $w^*>>0$, we have that the smooth minimal model $S$ of a generic
Koll\'ar surface is of general type with $K_{S}^2/e(S) \to 1$.
