Rudin's Real and Complex Analysis is an excellent book for several reasons. Most importantly, it manages to encompass a whole range of mathematics in one reasonably-sized volume. Furthermore, its problems are not mere extensions of the proofs given in the text or trivial applications of the results- many of the results are alternate proofs to major theorems or different theorems not proved. With that in mind, this book is not appropriate for a course where the instructor wants students to merely understand the theorems well enough to develop applications- the structure of the book is far better suited for a more theoretical course.
For example, the construction of Lebesgue measure is considered one of the most important topics in graduate analysis courses. After this construction, more abstract measures are developed, and then one proves the Riesz Representation Theorem for positive functionals later.
Conversely, Rudin develops a few basic topological tools, such as Urysohn's Theorem and a finite partition of unity, to construct the Radon measure needed in a sweeping proof of Riesz's Theorem. From this, results about regularity follow clearly, and the construction of Lebesgue measure involves little more than a routine check of its invariance properties.
Another example of where Rudin takes a more theoretical approach to provide a more elegant, yet less intuitive proof, is the Lebesgue-Radon-Nikodym theorem. Other books generally introduce signed measures with several examples, and use this result, along with properties of measures to derive the proof. On the other hand, since the first half of the book contains an intermission on Hilbert Space, Rudin uses the completeless of L^2 and the Riesz Representation Theorem for a more sweeping proof.
In the real analysis section, Rudin covers advanced topics generally not covered in a first course on measure theory. The chapters on differentiation and Fourier analysis are key examples of this. Rudin uses maximal functions to develop the Lebesgue Point theorem and results from complex analysis, and provides an incredibly thorough proof of the change-of-variables theorem. The ninth chapter, on Fourier transforms, relies heavily on convolutions, which are developed as a product of Fubini's theorem. This, in turn, is used to prove Plancherel's theorem and the uniqueness of Fourier transforms as a character homomorphism.
The tenth chapter, on basic complex analysis, essentially covers an entire undergraduate course on the subject, with added results based on a solid knowledge of topology on the plane. Once a solid foundation on the topic is laid, Rudin can develop more advanced topics from Harmonic analysis using general results from real analysis like the Hahn-Banach theorem and the Lebesgue Point theorem (for Poisson integrals).
Most of the basic results from the power series perspective are covered in the text, but while the geometric view is examined, it is still done in a very analytic, formula-based way that does not allow the reader to gain too much intuition. Nonetheless, all the basic results are covered, and Rudin uses these to develop the main theorems, such as the Mittag-Leffler and Weierstrass theorems on meromorphic functions, and the Monodromy Theorem and a modular function used to prove Picard's Little Theorem.
As an introductory text, even for advanced students, Rudin should probably be accompanied by more descriptive texts to develop better intuition. In fact, I would recommend Folland's Real Analysis and Ahlfors' Complex Analysis for self-study, because the problems are easier and one can learn better through those. With a good instructor, though, Rudin's text is concise and elegant enough to be both useful and enjoyable. It is also a good test to see how well one REALLY knows the subject.
