The inflationary universe scenario provides an appealing explanation for the riddles of why the observed universe is so flat and isotropic, and it neatly provides a causal mechanism for generating the primordial inhomogeneities which seed the large-scale structure we observe today. However, the 01igin of inflation is still unclear, mainly since we have no compelling evidence for a pe1iod of inflation from M-theory. Moreover, whether inflation actually occurs within a given theory is known to depend strongly on the initial conditions. These conceptual problems of inflation are the driving force behind the research presented in this thesis. We study inflation driven by higher de1ivative quantum effects and show how theories of initial conditions may provide observational discriminants between different inflationary scenarios. First we consider scalar field inflation and develop the techniques to calculate c01Telators of physical observables on a spacelike surface directly from the Euclidean no boundary path integral. We perturbatively evaluate the path integral for metric fluctuations around both regular and singular instanton saddle points. The real-space Euclidean two-point correlator is analytically continued into the Lorentzian universe, where it describes the quantum mechanical vacuum fluctuations of the graviton field in the state described by no boundary initial conditions. Unlike the usual approach to inflationary perturbations, the Euclidean path integral unambiguously specifies the graviton propagators with no additional assumptions. From our results for the two-point c01Telation functions we deduce predictions for the cosmic microwave background (CMB) anisotropies and compare this with what is observed today. It is shown that the long wavelength gravity waves would allow us to probe the initial state, if there were sufficient curvature left in the present-day universe. The lack of compelling theoretical evidence casts doubt on the claim that the potential energy of a scalar field could be the 01igin of an inflationary pe1iod in the early universe. It is therefore of the utmost importance to study realisations of inflation which do not require fundamental scalar fields that emerge more naturally from fundamental physics. In the second part of this thesis we revive an old idea of Starobinsky in which inflation is driven by the trace anomaly of a large number of conformally coupled matter fields. We argue that in contrast to scalar field inflation, trace anomaly driven inflation has a sound motivation in particle physics. We include in our action higher derivative counterterms, which arise naturally in the renormalisation of the conformal field theory (CFf). We discuss in detail how correlators of observables should be obtained from the path integral in a higher derivative theory. It is emphasised that the final boundary condition on the fields in the path integral is crucial to extract correct predictions for observations. Using the AdS/CFf correspondence, we present the first calculation of scalar and tensor metric propagators for trace anomaly inflation, taking full account of the back-reaction of matter fields. It is found that the primordial inhomogeneities are naturally suppressed, provided there are suffiently many matter fields, and that the higher derivative terms can play an important role in further reducing the fluctuations to the level we observe. The non-local part of the matter effective action has the effect of strongly suppressing fluctuations on small scales. This suggests that any small-scale modifications to four dimensional Einstein gravity would be unobservable in the CMB, since matter fields would dominate the graviton propagator at the scales at which such modifications might be expected to become important. To illustrate this point we consider a Randall-Sundrum (RS) brane world analogue of trace anomaly driven inflation, in which our universe is regarded as a domain wall in five dimensional anti-de Sitter (AdS) space with a large N conformal field theory living on the wall. It is shown that the effects of the matter fields dominate the RS corrections, rendering them unobservable. This result is probably not restricted to trace anomaly driven inflation, since it is a consequence of the presence of a large number of matter fields. We return to the problem of initial conditions for inflation at the end of the thesis. We provide evidence that the no boundary path integral predicts the Lorentzian inflationary phase in trace anomaly driven inflation to occur by semiclassical tunnelling via a four sphere instanton.