The paper examines intra-day share price volatility over the year 2000 for five market centres: the New York Stock Exchange, Nasdaq, the London Stock Exchange, Euronext Paris and Deutsche Börse. In each of these markets, we observe a U-shaped intra-day volatility pattern, a particularly sharp spike for the opening half hour, and a general level of intra-day volatility that is accentuated vis-à-vis volatility over longer differencing intervals, e.g. daily and weekly periods. We suggest that the volatility accentuation is attributable to spreads, market impact, price discovery and momentum trading - all of which are either trading costs or exist because of trading costs. Because the magnitude of trading costs depends in part on market design, we also suggest that a link exists between intra-day volatility and market structure, and that market quality/efficiency on both sides of the Atlantic could be improved.