In this paper, we consider the problem of designing Differentially Private\n(DP) algorithms for Stochastic Convex Optimization (SCO) on heavy-tailed data.\nThe irregularity of such data violates some key assumptions used in almost all\nexisting DP-SCO and DP-ERM methods, resulting in failure to provide the DP\nguarantees. To better understand this type of challenges, we provide in this\npaper a comprehensive study of DP-SCO under various settings. First, we\nconsider the case where the loss function is strongly convex and smooth. For\nthis case, we propose a method based on the sample-and-aggregate framework,\nwhich has an excess population risk of $\\tilde{O}(\\frac{d^3}{n\\epsilon^4})$\n(after omitting other factors), where $n$ is the sample size and $d$ is the\ndimensionality of the data. Then, we show that with some additional assumptions\non the loss functions, it is possible to reduce the \\textit{expected} excess\npopulation risk to $\\tilde{O}(\\frac{ d^2}{ n\\epsilon^2 })$. To lift these\nadditional conditions, we also provide a gradient smoothing and trimming based\nscheme to achieve excess population risks of $\\tilde{O}(\\frac{\nd^2}{n\\epsilon^2})$ and\n$\\tilde{O}(\\frac{d^\\frac{2}{3}}{(n\\epsilon^2)^\\frac{1}{3}})$ for strongly\nconvex and general convex loss functions, respectively, \\textit{with high\nprobability}. Experiments suggest that our algorithms can effectively deal with\nthe challenges caused by data irregularity.\n