Dynamical systems that are close to non-ergodic are characterised by the existence of subdomains or regions whose trajectories remain confined for long periods of time. A well-known technique for detecting such metastable subdomains is by considering eigenfunctions corresponding to large real eigenvalues of the Perron-Frobenius transfer operator. The focus of this thesis is to investigate the asymptotic behaviour of trajectories exiting regions obtained using such techniques. We regard the complement of the metastable region to be a ‘hole’, and show in Chapter 2 that an upper bound on the escape rate into the hole is determined by the corresponding eigenvalue of the Perron-Frobenius operator. The results are illustrated via examples by showing applications to uniformly expanding maps of the unit interval. In Chapter 3 we investigate a non-uniformly expanding map of the interval to show the existence of a conditionally invariant measure, and determine asymptotic behaviour of the corresponding escape rate. Furthermore, perturbing the map slightly in the slowly expanding region creates a spectral gap. This is often observed numerically when approximating the operator with schemes such as Ulam’s method. We investigate the asymptotic scaling of the spectral gap as the perturbation vanishes. In Chapter 4 we consider escape rate from random sets under the action of random dynamics and prove a result analogous to that of Chapter 2. We also show, under fairly weak assumptions, that in Oseledets subspaces Lyapunov exponents with respect to different norms are equal. The results are applied to Rychlik random dynamical systems. Finally, Chapter 5 deals with the main themes of the earlier chapters in the settings of deterministic and random shifts of finite type. There, we demonstrate methods to decompose shifts into complementary subshifts of large entropy. Much of the material in this thesis has either appeared in a scientific journal or has been submitted to one for publication.