We present a cubic planar hypohamiltonian graph on 70 vertices, improving the best known bound of 94 by Thomassen and derive some consequences concerning longest paths and cycles of planar 3-connected graphs. We also show that cubic planar hypohamiltonian graphs on n vertices exist for every even number n >= 86 and that cubic planar hypotraceable graphs exist on n vertices for every even number n >= 356, settling an open question of Holton and Sheehan.