An explicit recursion-theoretic definition of a random sequence or random set of natural numbers was given by Martin-Löf in 1966. Other approaches leading to the notions of n-randomness and weak n-randomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of n-random and weakly n-random sequences with an emphasis on the structure of their Turing degrees. After an introduction and summary, in Chapter II we present several equivalent definitions of n-randomness and weak n-randomness including a new definition in terms of a forcing relation analogous to the characterization of n-generic sequences in terms of Cohen forcing. We also prove that, as conjectured by Kurtz, weak nrandomness is indeed strictly weaker than n-randomness. Chapter III is concerned with intrinsic properties of n-random sequences. The main results are that an (n + 1)-random sequence A satisfies the condition A (n) ≡T A⊕0 (n) (strengthening a result due originally to Sacks) and that n-random sequences satisfy a number of strong independence properties, e.g., if A ⊕ B is n-random then A is n-random relative to B. It follows that any countable distributive lattice can be embedded