Biological invasions of non-indigenous species (NIS) are a severe and growing environmental problem that results primarily from introductions by humans. There are few tools for risk analysis of NIS introductions, most of which are insufficiently connected to ecological theory. This dissertation develops quantitative methods for forecasting species invasions and identifying intervention strategies. My approach is to develop relevant aspects of theoretical population biology and quantitative biogeography for representative species, while focusing on features that can be generalized. As a model, the spiny water flea Bythotrephes longimanus illustrates how understanding the population biology of target species can be crucial for effective management. I explore using stochastic population growth models to quantify the chance of biological invasion as a function of the number of organisms introduced. I find that seasonal fluctuations in environment conditions and the population biology of this species interact to create windows of invasion opportunity during which efforts at reducing introductions should be concentrated. I contribute to two areas of active research: population dynamics in (1) fluctuating environments and (2) open populations. Using Daphnia magna as a model species I test theoretical predictions with laboratory experiments. I follow these studies with a new model for establishment of parasitized populations. I predict that the chance of establishment is not likely to be greatly affected by parasitism. I use a genetic algorithm to identify the potential distributions of Eurasian ruffe and rainbow smelt in North America. I find that much of the Midwestern and Northeastern USA and Canada and some river systems along the Pacific coast may be invasible. I identify global hotspots for biological invasions from ballast water using data on global patterns of shipping traffic. I use a spatial interaction model to identify and evaluate possible interventions. I conclude that on-board ballast water treatment technologies will probably more effectively control invasions than ballast water treatment facilities. Finally, I consider whether the risk of invasion from ballast water could be managed by controlling the volume discharged. For sexually reproducing species this may be a feasible strategy if discharge occurs in open systems.