We prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. This extends earlier results by Hejhal [‘On the uniform equidistribution of long closed horocycles’, Loo-Keng Hua: a great mathematician of the twentieth century, Asian J. Math. 4 (2000) 839–853] and Strömbergsson [‘On the uniform equidistribution of long closed horocycles’, Duke Math. J. 123 (2004) 507–547] in dimension 2. Our proofs use spectral methods, and lead to precise estimates on the rate of convergence to equidistribution.