Regression is neccesarily based on numeric variables. There is no restriction given on the continuity of the variables. This means that the variables can be discrete, that's no problem.

The question is if and how one can construct a sensible numeric variable from some attribute that can take different nominal (non-numeric, catregorical) values. This is usually done by creating k-1 numeric indicator variables (that can take 0 or 1) for an attribute that can take k different values. Regression models based on such indicators as predictors are kown as "ANOVA models". Models that use such indicator variables as response variables are known as "logistic regression models" (depending on the number and order of the indicator variables one distingushes simple (binomial), multinominal and ordered logistic models.

In effect. the resulting transformed  data now  have measurement scale attributes as distinct from the original data labels.  Thank you immensely. Jocken for your explanation.

@Harold, the whole operation equally depends on the reactive variables within the set of the independent variables. There must an agreeable fact that a variable forms an integral part of the regressors. Secondly, on the linearity of the functions. Please look out for "Design and Analysis of Experiments, 5th Edition by DC Montgomery and Mathematical Analysis by JC Agunwamba. These may be of help. I wish you well. 

Kennedy, thank you for your contribution. My question  actually should have been restricted to linear regression only. Tha was what I had in mind  The response from Jocken was addressing the generalized issue of regression of which linear regression is a part.  Some regression models are non-linear.  I hope my comment is useful.