In this paper we develop two methods for completing partial latin squares and prove the following. Let $$A$$ A be a partial latin square of order $$nr$$ n r in which all non-empty cells occur in at most $$n-1$$ n - 1 $$r\times r$$ r × r squares. If $$t_1,\ldots , t_m$$ t 1 , … , t m are positive integers for which $$n\geqslant t_1^2+t_2^2+\cdots +t_m^2+1$$ n ⩾ t 1 2 + t 2 2 + ⋯ + t m 2 + 1 and if $$A$$ A is the union of $$m$$ m subsquares each with order $$rt_i$$ r t i , then $$A$$ A can be completed. We additionally show that if $$n\geqslant r+1$$ n ⩾ r + 1 and $$A$$ A is the union of $$n$$ n identical $$r\times r$$ r × r squares with disjoint rows and columns, then $$A$$ A can be completed. For smaller values of $$n$$ n we show that a completion does not always exist.