  An edge coloring of a graph $G$ is to color all the edges in the graph such
that adjacent edges receive different colors. It is acyclic if each cycle in
the graph receives at least three colors. Fiam{\v{c}}ik (1978) and Alon,
Sudakov and Zaks (2001) conjectured that every simple graph with maximum degree
$\Delta$ is acyclically edge $(\Delta + 2)$-colorable -- the well-known acyclic
edge coloring conjecture (AECC). Despite many major breakthroughs and minor
improvements, the conjecture remains open even for planar graphs. In this
paper, we prove that planar graphs are acyclically edge $(\Delta +
5)$-colorable. Our proof has two main steps: Using discharging methods, we
first show that every non-trivial planar graph must have one of the eight
groups of well characterized local structures; and then acyclically edge color
the graph using no more than $\Delta + 5$ colors by an induction on the number
of edges.
