Abstract 1 Introduction Many statistical and econometric models involve independence (or condi- tional independence) conditions that can be expressed via convolution. Ex- amples are independent errors, classical measurement error and Berkson er- ror, regressions involving data measured with these types of errors, common factor models and models that conditionally on some variables can be repre- sented in similar forms, such as a nonparametric panel data model with errors conditionally on observables independent of the idiosyncratic component. This paper provides a nonparametric analysis for several classes of mod- els, with cases such as classical measurement error, regression with errors in variables, and other models that may be represented in a form involving con- volution equations. The focus here is on conditions for existence of solutions, nonparametric identification and well-posedness in the space S ∗ of general- ized functions (tempered distributions). This space provides advantages over working in function spaces by relaxing assumptions and extending the results to include a wider variety of models, for example by not requiring existence of density. Classes of (generalized) functions for which solutions exist are defined; identification conditions, partial identification and its implications are discussed. Conditions for well-posedness are given and the related issues of plug-in estimation and regularization are examined. 2