Spatially variant versions of classic morphological operations are introduced. The probes used are locally variant on their domains. Range invariant morphological filters are introduced. A generalized Matheron Representation Theorem for these filters in terms of spatially variant erosions is proved. Locally variant openings and locally variant algebraic openings are introduced, and a generalized Matheron Representation Theorem for locally variant algebraic openings is proved. The theory presented is quite general and is valid for functions defined on subsets of any space X and with values in any ordered commutative group G for which the concept of least upper bound and greatest lower bound are meaningful. When G equals R, the real numbers, and X equals Rn the theory encompasses real signal processing (n equals 1), real image processing (n equals 2), and real multivariable signal processing (n > 2). When G equals Z, the integers, and X equals Zn the theory encompasses the digitized versions of each form of processing.