Litigation finance is an emerging investment activity in which third-party funders provide capital for legal disputes in exchange for a contingent return. Despite its rapid growth, little is known about how such financing reshapes litigation decisions, cost allocation, and risk exposure at the operational level. Using a large dataset of U.S. federal civil lawsuits, we first document stylized empirical facts on case duration, win rates, and awards, highlighting substantial and largely irreducible uncertainty in litigation outcomes and costs. Motivated by these findings, we model litigation as a stochastic investment problem in which legal effort affects both costs and the probability of success, and where agents differ in their risk preferences. We then extend the model to include a risk-neutral litigation funding company that offers contracts to law firms and plaintiffs. Our analysis yields three main insights. First, litigation finance expands the set of cases that are pursued by reallocating risk, particularly enabling high-risk cases that risk-averse law firms would otherwise decline. Second, uncertainty in case duration plays a central role: when duration risk is sufficiently low, more risk-averse law firms optimally invest more effort to reduce outcome variance, which can benefit plaintiffs but also highlights the potential impact of law firm consolidation on litigation intensity. Third, litigation finance systematically increases litigation spending, with the optimal contract assigning the funded effort entirely to the funder rather than sharing costs with the law firm. Overall, our results show how litigation finance alters effort incentives, case selection, and aggregate litigation activity through risk sharing and uncertainty management. While third-party funding can enhance access to justice, it may also increase court workloads, suggesting trade-offs that are central to the operational performance of the legal system.