We determine completely which graphical sequences $d_1 \geqq d_2 \geqq \cdots \geqq d_p $ with $d_1 - d_p = 1$ are planar graphical, and with a small number of exceptions determine the same result when $d_1 - d_p = 2$.We also give simple necessary conditions (in the form of upper bounds on $\sum\nolimits_{i = 1}^k {di} $) for a graphical sequence to be planar graphical. These conditions imply all known conditions of similar type, and often improve them.