Data $y_i$ are from a distribution within the exponential family, with mean $\mu_i$ and $g(\mu_i)=\textbf{x}'_i\boldsymbol{\beta}$ for some link function, $g$.\\

The exponential family includes distributions such as the Gaussian, binomial, Poisson, and gamma (and thus exponential and chi-squared).\\

The link functions are typically
\begin{itemize}
\item identity (with the Gaussian)
\item log (with the Poisson and the gamma)
\item logistic (with the binomial)
\end{itemize}

A likelihood function can be set up for each of these models and the parameters can be estimated using maximum likelihood.

The glm package in R has options to estimate parameters in these models.