The real problem with libertarianism:
It doesn't solve your problems, or perhaps more accurately, it offers no internal solutions for them.
The remarkable thing about those who complain about libertarianism is that the alternatives they promote -also- don't solve your problems; the problems they promote as inherent to libertarianism are invariably problems which were or are encountered in governmental systems which definitively were or are -not- libertarian.
First and foremost, a libertarian government IS A GOVERNMENT. It's a limited government (the precise limits vary by libertarian - some favor social safety nets, some, like myself, favor a minimal government whose domain is limited to that of violence), but it is a government. Most libertarians hold that government should have taxing power - those who do not favor arrangements such as lotteries or other voluntary fundraising mechanisms.
It's not lawlessness; indeed, libertarian principles are rooted in the concept of rule of law, and hold as a prime principle that the government must be subject to the law even as it is its enforcer, which is why their favored governments are constitutional. (A constitution being the set of laws which dictate how the government must behave.)
We don't favor rich people, we just don't think government should hold them in disfavor in order to provide some sort of cosmic balance for the way they've been elsewise favored; libertarians believe justice must be blind, and any justice which isn't isn't justice at all.
In general, we're not particularly fond of corporations. A lot of us are accustomed to defending them from baseless assault, a habit which can lead to defending them from reasonable criticism, but most of us dislike the way they indemnify the people who run them against tort, a legal privilege which runs against the rule of law.
As a rule of thumb libertarians oppose collectivization. This leads us to oppose both racism and affirmative action, both of which treat people as collectives rather than individuals. Statistics are frequently devoid of information which is personally meaningful to us, although we're not opposed to using them in arguments with people for whom they are meaningful.
Many, but not all, of us regard trade in something the same manner that others regard religious beliefs; a personal matter. No trade is fundamentally different than two boys in a cafeteria exchanging trading cards, or anything else of mutual value. A company is as a church, a collection of people sharing similar purpose; and as the freedom for the individual to pray in his home is no different from his freedom to collect with others to pray together (or to abstain from such activities), so to is a group of people working to mutual benefit no different from those boys.
Contracts are enforceable as a matter of ownership, a principle common law agrees with us on. (Common law on contracts can only enforce by principles of ownership; a contract without mutual trade of value is unenforceable. For an example of how and why this matters, if you sign a contract to let an oil company dig a retention pool on your property in exchange for them filling it in later and planting some trees, and they refuse to fill it in and plant the trees, common law provides only the difference in property value, not the cost of doing what was agreed upon.)
Because there's a variety of positions of libertarian, the reasons for diverging from other political philosophies also vary. Pragmatic libertarians believe the best government for (X criteria) is a smaller government; principle libertarians are generally closer to the minarchistic side, and most frequently abide by the non-aggression principle.
Many anarchists also cite the non-aggression principle as a reason for their political position; libertarians and anarchists are at odds over who better abides by it. Anarchists hold that forcing people to abide under a specific government is a violation of the non-aggression principle; libertarians hold that anarchistic modes of governance effectively make the non-aggression principle optional (or in some modes of anarchy, negotiable), and so violate it by default. I don't personally find that there's a clear winner there, and default to libertarianism on pragmatic grounds. (Granted, my personal conviction that anarchy is untenable depends upon armchair logic, but their conviction that it is tenable is no different in that regard.)
Some libertarians believe the right rules can make a libertarian government permanent; I personally hold that no governmental system is permanent, and regard the slow shift to a social democracy unavoidable, albeit correctable. (I hold that there is entropic decay in any government which will eventually lead to its destruction; entropy can be countered by a constant effort by a determined populace, but I hold in addition that the societal structures which produce this effort themselves are also subject to entropic decay; you can add as many layers of protection as you want, they'll each eventually collapse in turn.)
Libertarians don't hold particularly interesting views, once you look at their reasons. Just different ones.