A renewal equation is an integral equation with respect to an unknown function. Such equations have found numerous applications in many areas of applied probability. In particular, defective renewal equations are found very frequently and are important mathematical tools in reliability theory, demography, queuing theory and in actuarial science. In this thesis it is studied the behavior of the solution of a defective renewal equation. Specifically, they are considered conditions that ensure the existence of monotonicity for this function. Also, generalizing the methods of Cai and Garrido (1998), bounds are constructed and it is studied the asymptotic behavior for this solution. Then, the thesis focuses on a specific renewal equation, the solution of which is the kth moment of the ruin time, ψk(u), in the classical model of risk theory. Considering this quantity as a function of the initial reserve u in the model, bounds are presented and examining the existence of monotonicity for the function ψk(u). Also, it is studied the asymptotic behavior of this function in the case that the distribution of claims has light, medium or heavy tail. Finally, it is investigated the existence of aging properties of the distribution of the ruin time. Apart from the classical model that mentioned above, many of the results are generalized for the model introduced by Gerber (1970) and include a diffusion term.