Maximal functions such as the Hardy-Littlewood maximal function play a fundamental role in various areas of mathematical analysis. Since Stein’s seminal work on the spherical maximal function in the 1970s, there have been efforts to prove L^p boundedness of the maximal functions defined by averages over submanifolds. This talk concerns the maximal functions defined by averages over curves. In two dimensions, the celebrated Bourgain’s circular maximal theorem establishes L^p boundedness. In higher dimensions, no results have been known until recently. We prove that the maximal function along a nondegenerate space curve is bounded on L^p if and only if p>3. It settles a long-standing open problem that has remained open since the 1980s. We also prove the existence of nontrivial L^p bound in any dimension.