This thesis presents research in the area of Solar System dynamics, specifically the long-term orbital evolution of small Solar System bodies. The stable lifetimes of objects in the inner Edgeworth-Kuiper Belt (EKB) were investigated. From a plausible initial model distribution, the number of objects which should survive over the age of the Solar System was found. The stable lifetimes of actual objects were also evaluated. It was fou~d that there are more objects present in this unstable region than is possible due to gravitational factors alone. Two potential processes to move objects into the inner EKB were suggested and investigated: interactions with as yet undiscovered large objects, and the effects of collisions in another part of the belt. The former process did not have a large effect on the population of the region, whilst the latter process did increase the time objects spent in the region. Observational studies of one particularly unstable object provided evidence that it has undergone one or more collisions, . supporting the idea of a collisional mechanism. Trojan asteroids of Jupiter were investigated to determine how useful the Lyapunov Characteristic Exponent (LCE) is in predicting a Trojan's stability. It was found to be of some limited usefulness. The dynamical evolution of the Kappa Cygnid meteoroid stream was studied. It was found that the stream can be divided into two substreams. Two near-Earth objects which are potential parents of one of these were found . . The use of the mean and median orbits to represent meteoroid streams were compared, and it was found that the mean provided a more accurate represent ation. The lifetimes of meteorite streams (streams of large particles capable of surviving the Earth's atmosphere and becoming meteorites) were studied. It was found that streams can exist in the inner Solar System for 30 Myr.