  The fact that closed timelike curves (CTCs) are permitted by general
relativity raises the question as to how quantum systems behave when time
travel to the past occurs. Research into answering this question by utilising
the quantum circuit formalism has given rise to two theories: Deutschian-CTCs
(D-CTCs) and "postselected" CTCs (P-CTCs). In this paper the quantum circuit
approach is thoroughly reviewed, and the strengths and shortcomings of D-CTCs
and P-CTCs are presented in view of their non-linearity and time travel
paradoxes. In particular, the "equivalent circuit model"---which aims to make
equivalent predictions to D-CTCs, while avoiding some of the difficulties of
the original theory---is shown to contain errors. The discussion of D-CTCs and
P-CTCs is used to motivate an analysis of the features one might require of a
theory of quantum time travel, following which two overlapping classes of new
theories are identified. One such theory, the theory of "transition
probability" CTCs (T-CTCs), is fully developed. The theory of T-CTCs is shown
not to have certain undesirable features---such as time travel paradoxes, the
ability to distinguish non-orthogonal states with certainty, and the ability to
clone or delete arbitrary pure states---that are present with D-CTCs and
P-CTCs. The problems with non-linear extensions to quantum mechanics are
discussed in relation to the interpretation of these theories, and the physical
motivations of all three theories are discussed and compared.

Quantum Physics;General Relativity and Quantum Cosmology