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Matching markets with historical data are abundant in many applications, e.g., matching candidates to jobs in hiring, workers to tasks in crowdsourcing markets, and jobs to servers in cloud services. In all these applications, a match consumes one or more shared and limited resources and the goal is to best utilize these to maximize a global objective. Additionally, one often has historical data and hence some statistics (usually first-order moments) of the arriving agents (e.g., candidates, workers, and jobs) can be learnt. To model these scenarios, we propose a unifying framework, called Multi- Budgeted Online Assignment with Known Adversarial Distributions. In this model,we have a set of offline servers with different deadlines and a set of online job types. At each time, a job of type j arrives. Assigning this job to a server i yields a profit w(i, j) while consuming a(i,j) -- a vector lying in [0, 1]^K -- quantities of distinct resources. The goal is to design an (online) assignment policy that maximizes the total expected profit without violating the (hard) budget constraint. We propose and theoretically analyze two linear programming (LP) based algorithms which are almost optimal among all LP-based approaches. We also propose several heuristics adapted from our algorithms and compare them to other LP-agnostic algorithms using both synthetic as well as real-time cloud scheduling and public safety datasets. Experimental results show that our proposed algorithms are effective and significantly out-perform the baselines. Moreover, we show empirically the trade-off between fairness and efficiency of our algorithms which does well even on fairness metrics without explicitly optimizing for it.