An \(m\times n\) matrix \(A\) is called \(TP_k\) (totally positive k-by-k) if every minor of size at most \(k\) is positive. If \(k = \min\{m, n\}\), the matrix is simply called \(TP\) (totally positive). The authors generalize these notions to ``shapes'' as a generalization of matrices. They also characterize the relationship between positivity of ``contiguous'' minors and all minors for general shapes and for certain special types of shapes. They introduce the notions of barriers and thickness and apply their results to the \(TP_k\)-completion problem and \(TP_k\)-completable patterns. Finally the authors give an explicit combinatorial condition for a large class of patterns to be \(TP_2\)-completable.