In this paper, we define and compare four new measures of graph irregularity. We use these measures to prove upper bounds for the chromatic number and the Colin de Verdière parameter. We also strengthen the concise Turán theorem for irregular graphs and investigate to what extent Turán's theorem can be similarly strengthened for generalized r-partite graphs. We conclude by relating these new measures to the Randić index and using the measures to devise new normalised indices of network heterogeneity.