Nonstandard Analysis arose from the observation that there exists a model of the reals with infinitesimals, and thus infinitesimal reasoning can be made equivalent to standard (epsilon-delta) arguments. Somewhat ironically, however, Abraham Robinson's original model-theoretic approach was avoided like the plague. His book's first chapter, Tools From Logic, was known for its notorious difficulty, and mathematicians quickly developed alternatives like Internal Set Theory or hybrid approaches.
This paper attempts to show that a clear, concise introduction in the context of Model Theory is now possible. In the several decades that have passed, developments have been made not only in NSA (and typesetting), but in Model Theory itself, allowing for a more streamlined portrayal (our own Tools, all the way to the construction of the saturated model, takes no more than nine pages). Furthermore, we believe this method makes it most apparent why Nonstandard Analysis is designed the way it is, something which is often lost in the modern treatment.
Even if this particular introduction is missing something to your liking, we hope that it at least demonstrates the possibility of an acceptable one. We also hope that the reader finds it illustrious and insightful.
Nonstandard Analysis from a Model-Theoretic Perspective
Follow-up: The Hyperreals (draft)
Please contact me at saichu dot math at gmail dot com with any suggestions.