This thesis combines computability theory and various notions of fractal dimension, mainly Hausdorff dimension. An algorithmic approach to Hausdorff measures makes it possible to define the Hausdorff dimension of individual points instead of sets in a metric space. This idea was first realized by Lutz (2000). Working in the Cantor space of all infinite binary sequences, we study the theory of Hausdorff and other dimensions for individual sequences. After giving an overview over the classical theory of fractal dimension in Cantor space, we develop the theory of effective Hausdorff dimension and its variants systematically. Our presentation is inspired by the approach to algorithmic information theory developed by Kolmogorov and his students. We are able to give a new and much easier proof of a central result of the effective theory: Effective Hausdorff dimension coincides with the lower asymptotic algorithmic entropy, defined in terms of Kolmogorov complexity. Besides, we prove a general theorem on the behavior of effective dimension under r-expansive mappings, which can be seen as a generalization of Holder mappings in Cantor space. Furthermore, we study the connections between other notions of effective fractal dimension and algorithmic entropy. Besides, we are able to show that the set of sequences of effective Hausdorff dimension s has Hausdorff dimension s and infinite s-dimensional Hausdorff measure (for every 0