We study the problem of learning a hypergraph via edge detecting queries. In this problem, a learner queries subsets of vertices of a hidden hypergraph and observes whether these subsets contain an edge or not. In general, learning a hypergraph with m edges of maximum size d requires Omega((2m/d)^{d/2}) queries. In this paper, we aim to identify families of hypergraphs that can be learned without suffering from a query complexity that grows exponentially in the size of the edges. We show that hypermatchings and low-degree near-uniform hypergraphs with n vertices are learnable with poly(n) queries. For learning hypermatchings (hypergraphs of maximum degree Delta = 1), we give an O(log^3 n)-round algorithm with O(n log^5 n) queries. We complement this upper bound by showing that there are no algorithms with poly(n) queries that learn hypermatchings in o(log log n) adaptive rounds. For hypergraphs with maximum degree Delta and edge size ratio rho, we give a non-adaptive algorithm with O((2n)^{rho Delta+1} log^2 n) queries. To the best of our knowledge, these are the first algorithms with poly(n, m) query complexity for learning non-trivial families of hypergraphs that have a super-constant number of edges of arbitrarily large size.