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We study an N=1 two-dimensional non-linear sigma model with boundariesrepresenting, e.g., a gauge fixed open string. We describe the full set ofboundary conditions compatible with N=1 superconformal symmetry. The problem isanalyzed in two different ways: by studying requirements for invariance of theaction, and by studying the conserved supercurrent. We present the target spaceinterpretation of these results, and identify the appearance of partiallyintegrable almost product structures.Comment: 24 pages, LaTeX. v2: minor clarification in section 3.2, reference  added. v3: section added on N=2 supersymmetry. v4: typos corrected, published  versio

N = 1 Supersymmetric Sigma Model with Boundaries, I