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I want to know what is the best way to analyze a data set where my response variable is count data and my explanatory variables are continuous variables. All my variables are not normally distributed. Are GLMs a good option?

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They are. You may want to look at Poisson regression (in R: glm(..., family=poisson, ...)) or, if you have overdispersion, Negbin regression or, if you have "too many" zeros, ZIP regression (Zero-Inflated Poisson).

Whether the predictors are normally distributed does not matter. (Except for analyses of influential data points.) What you probably have in mind is whether residuals are normally distributed. This is an important assumption in Ordinary Least Squares - more specifically: for inference in OLS. However, your data are counts, so residuals will not be normal and you are not thinking about OLS, anyway.

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it might be good to clarify your second paragraph so that it is clear it applies only to the post's idea about normality of variables. – Peter Flom Jan 17 at 19:45
@PeterFlom: good point, thanks! I edited my question. – Stephan Kolassa Jan 17 at 19:50
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For a non-normal GLM, the distribution of residuals do not matter, I have never heard that a Poisson GLM for example has to have normal errors. I think that examining overdispersion and general model fit are better cues to which model to use. Negative binomial can be implemented in glm.nb() in the R MASS library, and ZIP can be done using the zeroinfl() function in the pscl library in R

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"For a non-normal GLM, the distribution of residuals do not matter" - this statement is false, the distribution of residuals and thus the error structure is strictly defined by the GLM family: poisson, binomial, ... – Tomas Jan 17 at 19:11
But it is not strictly normal – Corey Sparks Jan 17 at 19:38

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