This paper studies models of processes generating censored outcomes with endogenous explanatory variables and instrumental variable restrictions. Tobit-type left censoring at zero is the primary focus in the exposition. The models studied here are unrestrictive relative to others widely used in practice, so they are relatively robust to misspecification. The models do not specify the process determining endogenous explanatory variables and they do not embody restrictions justifying control function approaches. The models can be partially or point identifying. Identified sets are characterized and it is shown how inference can be performed on scalar functions of partially identified parameters when exogenous variables have rich support. In an application using data on UK household tobacco expenditures inference is conducted on the coefficient of an endogenous total expenditure variable with and without a Gaussian distributional restriction on the unobservable and compared with the results obtained using a point identifying complete triangular model.