We study percolation in the hyperbolic plane
H
2
\mathbb {H}^2
and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and Poisson-Voronoi-Bernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase,
p
∈
(
0
,
p
c
]
p\in (0,p_c]
, there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase,
p
∈
(
p
c
,
p
u
)
p\in (p_c,p_u)
, there are infinitely many unbounded clusters for the process and for the dual process. In the third phase,
p
∈
[
p
u
,
1
)
p\in [p_u,1)
, there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of
p
c
p_c
in the Poisson-Voronoi-Bernoulli percolation process on the intensity of the underlying Poisson process.