We introduce the non-negative matrix factor 2-D deconvolution (NMF2D) model, which decomposes a matrix into a 2-dimensional convolution of two factor matrices. This model is an extension of the non-negative matrix factor deconvolution (NMFD) recently introduced by Smaragdis (2004). We derive and prove the convergence of two algorithms for NMF2D based on minimizing the squared error and the Kullback-Leibler divergence respectively. Next, we introduce a sparse non-negative matrix factor 2-D deconvolution model that gives easy interpretable decompositions and devise two algorithms for computing this form of factorization. The developed algorithms have been used for source separation and music transcription.