How do voters think about political representation? Recent theoretical work suggests that representation is best described as a fundamentally multidimensional concept, but existing empirical work is limited to evaluating either the ideological or the demographic congruence between voters and politicians. In this paper, we introduce new survey instruments to expand our understanding of how voters think about their political representatives. First, we describe a new survey battery which is straightforward to implement in existing surveys; which applies to different political systems; and which captures voter preferences over six distinct dimensions of representation. Second, we use a conjoint design to evaluate the relative importance of these dimensions in the minds of voters when they consider the representational merits of different politicians. Applying our approach to the UK, the US, and Germany, we find that voters' representational demands differ systematically across political systems, and that individual voter characteristics are also strongly predictive of preferences over representation. A core implication of our findings is that voters do not have a uniform desire for 'more' or 'less' representation, but rather that different voters favour different aspects of representation in different political contexts.