Understanding human mobility is crucial for a broad range of applications from disease prediction to communication networks. Most efforts on studying human mobility have so far used private and low resolution data, such as call data records. Here, we propose Twitter as a proxy for human mobility, as it relies on publicly available data and provides high resolution positioning when users opt to geotag their tweets with their current location. We analyse a Twitter dataset with more than six million geotagged tweets posted in Australia, and we demonstrate that Twitter can be a reliable source for studying human mobility patterns. Our analysis shows that geotagged tweets can capture rich features of human mobility, such as the diversity of movement orbits among individuals and of movements within and between cities. We also find that short- and long-distance movers both spend most of their time in large metropolitan areas, in contrast with intermediate-distance movers' movements, reflecting the impact of different modes of travel. Our study provides solid evidence that Twitter can indeed be a useful proxy for tracking and predicting human movement.