This paper focuses on fingering instability of a thin liquid film flowing down a vertical uniformly heated/cooled cylinder under the influence of gravity. Thin film model is applied to derive the evolution equation and linear stability analysis in three dimensional setting is investigated. According to the evolution equation, the height of the thin film flowing down a cylinder depends on two parameters, the Marangoni number and the dimensionless cylinder radius. Travelling wave solutions are constructed, and the influence of the Marangoni number, the thickness of the precursor film and the radius of the cylinder are taken into consideration. Fully three dimensional simulations are also given based on finite element method with the help of open source software Freefem++. Numerical simulations demonstrate that the Marangoni effect enhances long-wave instability while the thickness of the precursor film suppresses fingering instability. The dimensionless radius has a significant destabilizing influence on the flow while the scaled radius R < 1. Fully 3D simulations match very well with the linear stability analysis, showing that the fingers grow faster with positive Marangoni effect when the fingers grow slower with negative Marangoni effect. (C) 2019 Elsevier Ltd. All rights reserved.