We consider the problem of allocating objects to agents when the objects have minimum quotas. There exist many realworld settings where minimum quotas are relevant. For example, in a hospital-resident matching problem, unconstrained matching may produce too few assignments to a rural hospital. Surprisingly, almost 50 years have passed after the seminal work by Gale and Shapley, no existing mechanism can guarantee minimum quotas so far; we did not know how to guarantee that a rural hospital has at least one resident. In this paper, we propose mechanisms that can satisfy minimum quotas as well as standard maximum quotas. More specifically, we propose extended seat (ES) and multi-stage (MS) mechanisms modeled after the well-known deferredacceptance (DA) and top trading cycles (TTC) mechanisms. Our proposed mechanisms are all strategy-proof, but a tradeoff exists between the DA and TTC based mechanisms regarding Pareto efficiency and elimination of justified envy. In addition, there exist a tradeoff between ES and MS mechanisms depending on the size of minimum quotas.