The maximum constraint satisfaction problem (Max CSP) is the following computational problem: an instance is a finite collection of constraints on a set of variables, and the goal is to assign values to the variables that maximises the number of satisfied constraints. Max CSP captures many well-known problems (such as Max k -SAT and Max Cut) and so is NP-hard in general. It is natural to study how restrictions on the allowed constraint types (or constraint language) affect the complexity and approximability of Max CSP. All constraint languages, for which the CSP problem (i.e., the problem of deciding whether all constraints in an instance can be simultaneously satisfied) is currently known to be NP-hard, have a certain algebraic property, and it has been conjectured that CSP problems are tractable for all other constraint languages. We prove that any constraint language with this algebraic property makes Max CSP hard at gap location 1, thus ruling out the existence of a polynomial-time approximation scheme for such problems. We then apply this result to Max CSP restricted to a single constraint type. We show that, unless P = NP, such problems either are trivial or else do not admit polynomial-time approximation schemes. All our hardness results hold even if the number of occurrences of each variable is bounded by a constant.