TITLE: (infinity,1)-categories directly from model categories
QUESTION [9 upvotes]: Edit & Note: I'm declaring a convention here because I don't feel like trying to fix this in a bunch of spots:  If I said model category and it doesn't make sense, I meant a model-category "model" of an (infinity,1)-category.  Also, "model" in quotes means the English word model, whereas without quotes it has do do with model categories.
At the very beginning of Lurie's higher topos theory, he mentions that there is a theory of $(\infty,1)$-categories that can be directly constructed by using model categories.  
What I'd like to know is: 
Where can I find related papers (Lurie mentions two books that are not available for download)?
How dependent on quasicategories is the theory developed in HTT?  Can the important results be proven for these $(\infty,1)$-model-categories by proving some sort of equivalence (not equivalence of categories, but some weaker kind of equivalence) to the theory of quasicategories?
When would we want to use quasicategories rather than these more abstract model categories?  
And also, conversely, when would we want to look at model categories rather than quasicategories?  
Does one subsume the other?  Are there disadvantages to the model category construction just because it requires you to have all of the machinery of model categories? Are quasicategories better in every way?
The only "models" of infinity categories that I'm familiar with are the ones presented in HTT.

REPLY [6 votes]: Here is an extended comment regarding Charles answer, including some more references.
In this paper, Theorem 2.5.9, it is shown that every model category (not necessarily a combinatorial one) has all limits and colimits. However, it is not hard to find examples of model categories whose underlying $\infty$-categories are neither presentable nor co-presentable. For instance, Isaksen's strict model structure on pro-simplicial sets. It is shown in the paper mentioned above that the underlying $\infty$-category of this model category is the pro category of spaces considered in Lurie's "Higher Topos Theory" Definition 7.1.6.1. The pro category of a large cocomplete and finitely complete $\infty$-category is complete and cocomplete but neither presentable nor copresentable. 
In this paper, Proposition 1.5.1, it is shown that any Quillen pair between model categories (not necessarily combinatorial ones) gives rise to an adjoint pair of $\infty$-categories. 
It seems plausible to me that the (underlying $\infty$-category of the) relative category of model categories and left Quillen functors between them, with weak equivalences taken to be the Quillen equivalences is equivalent to the $\infty$-category of complete and cocomplete $\infty$-categories and left adjoints between them. Restricting to combinatorial model categories would correspond to restricting to presentable $\infty$-categories in the image. The latter statement is probably already proven, whereas I am pretty sure the former is not.