In this paper, we introduce some insurance loss models within a Markov process framework. We start by introducing a univariate risk process in which the claims arrives according to a Markovian arrival process (MAP) and present formulas for the Laplace transform and the moments of the present value of the aggregate losses that have occurred during a fixed time period (0; t). Then to model the multi-type of losses incurred by an insurance company, we introduce a multivariate aggregate loss model based on Marked Markovian arrival processes (MMAP). This multivariate risk model may considers different types of losses, allowing dependencies among the claim frequencies, among the claim severities, as well as between claim frequencies and claim severities. We give the joint Laplace transform and the joints moments of the present value of the aggregate losses that have occurred during a fixed time period (0; t). To measure the risk level of the loss processes, we consider ruin probability related quantities. To this end, or the uinivariate model, we provide formulas and computation procedures for the probability of ruin and the deficit at ruin; for the multivarite model, we provide formulas and computation methods for the probabilities of ruin due to different types of claims. We provide numerical examples to illustrate the usefulness of the model.