Well, we could go on forever listing various critiques from both the Left and the Right, so I’ll just cover a few and maybe other people will stop by and list more.
There are lots of criticisms of liberalism from the Marxist and socialist corners. We could be here all day listing them, so I’ll just mention one that hits at the heart of liberalism, which is freedom. The charge is that the kind of freedom valued by liberalism is a very limited kind of freedom, mainly a sort of freedom to be an actor in capitalism. Think of this part from the Communist Manifesto:
And the abolition of this state of things is called by the bourgeois, abolition of individuality and freedom! And rightly so. The abolition of bourgeois individuality, bourgeois independence, and bourgeois freedom is undoubtedly aimed at. By freedom is meant, under the present bourgeois conditions of production, free trade, free selling and buying. But if selling and buying disappears, free selling and buying disappears also. This talk about free selling and buying, and all the other “brave words” of our bourgeois about freedom in general, have a meaning, if any, only in contrast with restricted selling and buying, with the fettered traders of the Middle Ages, but have no meaning when opposed to the Communistic abolition of buying and selling, of the bourgeois conditions of production, and of the bourgeoisie itself.
So what liberalism conceives of as restrictions on freedom, like for instance the sorts of measures that might be in place in a communist society, are in fact only restrictions on a warped notion of freedom that depends on the conditions of capitalism for its attractiveness.
Again, there are lots of criticisms that fall under this broad umbrella, and I’ll just mention one. Liberalism is very concerned with autonomy and autonomous choices, but many feminist understandings of autonomy move away from the traditional liberal conception of the isolated individual to a notion of autonomy that sees it as an inherently relational property which arises out of people being situated in certain ways in society. If this is our understanding of autonomy, much of classical liberalism makes no sense: for instance, the social contract model of the state, according to which consent from each person is what legitimizes the state, breaks down, because we can’t coherently speak of consent or any other function of an individual’s autonomy until we already have on the table the structure of society. If that structure includes the state, and presumably it does, then the state is somehow prior to the people consenting to it, which is bad news for liberals. We could draw links here to Hegel and communitarianism, which will come up later when we look at the Right.
In The Racial Contract Mills argues that social contract theory is predicated on white supremacy and that all the ostensibly color-blind theories of liberalism built around it are in fact just reifications of racism. Mills actually thinks liberalism can be saved in the form of what he calls “black radical liberalism” (this is a somewhat recent development - see here for instance) but one might disagree with him, and even if we agreed, I think this still counts as a critique of liberalism, right?
By this I don’t mean actual philosophical pragmatism but rather the view that sometimes, liberalism isn’t tenable simply because respect for individual rights will lead to consequences too dire to accept. So, this is just a straightforward consequentialist argument: the ends justify the means, and sometimes the ends will require adopting means other than liberalism. So for instance Arneson has advocated for an instrumentalist defense of democracy (see here) according to which there is a right to democratic participation only insofar as democracy is going to generate good results in that society, and if this isn’t the case, then there’s no such right (see also his article “On the Supposed Right to a Democratic Say”). We might call these people fair-weather liberals. They have something in common with the communitarians, insofar as the character of the society in question helps decide whether various facets of liberalism are appropriate.
This is what has its roots in Hegel, and we can see it in people like Taylor and Sandel, cited here. The broadest possible way of describing what’s going on here is that there are different principles fit for different societies, depending on the character of those societies. So if a society has illiberal traditions, it typically doesn’t make sense to come in with a liberal steamroller and tell them that they’re doing everything wrong and that they have to change. We might think morality simply doesn’t work this way, either because there’s no such universal morality in the first place, or because the way morality works requires it getting a certain foothold in the individual’s life in a way that makes sense to that individual and not all people in all societies will be amenable to liberalism, or whatever. Another facet of this critique (especially from Sandel) echoes the feminist point above: the idea is that it makes no sense to conceive of the individual outside the context of their society, and to talk about the rights and choices of that individual in any meaningful sense.
If you want any more detail on any of these answers, let me know. I’m not sure how much you know about liberalism: I’ve assumed a fair amount of knowledge on your part, and thus left out much of the details in terms of what parts of liberalism these critiques are attacking and how they hurt, insofar as they succeed. I’d be happy to fill that out, or anything else that needs filling out.