Let me begin with what looks like a joke. According to a Bourbaki member, the following conversation occurred during a meeting dedicated to polishing the but-last version of an Algebra Bourbaki volume:


(a Bourbaki member) Why not state explicitly that the coefficients of cyclotomic polynomials are $0,\pm 1$ ?


(another member) Because it's false.


Here is what I am aware: if $n$ has at most two distinct odd prime factors, then the coefficients of $\Phi\_n(X)$ are $0,\pm1$ (Migotti, 1883). In other words, this holds true for $n=2^mp^kq^\ell$, where $p,q$ are primes. On the other hand, it is false for $n=105=3\cdot5\cdot7$, because the coefficient of $X^7$ (or of $X^{41}$ as well) is $-2$.



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> My question is whether there is a complete characterization of those $n$ for which the coefficients of $\Phi\_n(X)$ are $0,\pm1$ ? If not, are there other infinite lists of cyclotomic polynomials with this property?
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Cyclotomic polynomials with coefficients $0,\pm1$
There are other families, but there is by no means a complete characterization known. 
Even for products of three primes there is no complete answer known. A relevant keyword is flat cyclotomic polynomial. Some results to give a flavor of the problem. 


The follwing is due to N. Kaplan (from some years ago, Journal of Number Theory):



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> Let $p,q,r$ be primes (strictly increasing in size), if $r$ is $1$ or $-1$ modulo $pq$ then the coefficients of the $pqr$ cyclotomic pollynomial are only $0,1, -1$.
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He also has some periodicity result that allows constructions: let $n$ be an integer and $s,t$ primes strictly greater $n$, and congruent modulo $n$ then the set of coefficients of the $nt$ and $ns$ cyclotomic polynomial coincide. 


In 2010 S. Elder obtained further results in this direction. 



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> Let $p,q,r$ be primes (strictly increasing in size), if $r$ is $2$ or $-2$ modulo $pq$ then the coefficients of the $pqr$ cyclotomic polynomial are only $0,1, -1$ if and only if $q$ is $1$ modulo $p$. 
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Elder also has some additional results for products of three primes, and also for products of four and five primes; [see this presentation of Elder](http://www.coloradomath.org/handouts/flat_cyclotomic.pdf) where also the above mentioned results can be found.


