Before revision:  Based on the robust existence of quadratic variation and a certain topology which is induced by Vovk's outer measure, model-free It\^o integration is developed on the space of continuous paths, of non-negative c\`adl\`ag paths and of c\`adl\`ag paths with mildly restricted jumps .
After revision:  Based on the robust existence of the quadratic variation, a model-free It\^o integration is developed on the space of continuous paths, of non-negative c\`adl\`ag paths and of c\`adl\`ag paths with mildly restricted jumps .