We solve the spectral synthesis problem for exponential systems on an interval. Namely, we prove that any complete and minimal system of exponentials {eiλnt}n∈N in L2(−a,a) is hereditarily complete up to a one-dimensional defect. This means that for any partition N=N1∪N2 of the index set, the orthogonal complement to the system {eiλnt}n∈N1∪{en′}n∈N2, where {en′} is the system biorthogonal to {eiλnt}, is at most one-dimensional. However, this one-dimensional defect is possible and, thus, there exist nonhereditarily complete exponential systems. Analogous results are obtained for systems of reproducing kernels in de Branges spaces. For a wide class of de Branges spaces we construct nonhereditarily complete systems of reproducing kernels, thus answering a question posed by N. Nikolski.