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We introduce the logic FOCN(P) which extends first-order logic by countingand by numerical predicates from a set P, and which can be viewed as a naturalgeneralisation of various counting logics that have been studied in theliterature.  We obtain a locality result showing that every FOCN(P)-formula can betransformed into a formula in Hanf normal form that is equivalent on all finitestructures of degree at most d. A formula is in Hanf normal form if it is aBoolean combination of formulas describing the neighbourhood around its tupleof free variables and arithmetic sentences with predicates from P over atomicstatements describing the number of realisations of a type with a singlecentre. The transformation into Hanf normal form can be achieved in timeelementary in $d$ and the size of the input formula. From this locality result,we infer the following applications: (*) The Hanf-locality rank of first-orderformulas of bounded quantifier alternation depth only grows polynomially withthe formula size. (*) The model checking problem for the fragment FOC(P) ofFOCN(P) on structures of bounded degree is fixed-parameter tractable (withelementary parameter dependence). (*) The query evaluation problem for fixedqueries from FOC(P) over fully dynamic databases of degree at most d can besolved efficiently: there is a dynamic algorithm that can enumerate the tuplesin the query result with constant delay, and that allows to compute the size ofthe query result and to test if a given tuple belongs to the query resultwithin constant time after every database update.

First-Order Logic with Counting: At Least, Weak Hanf Normal Forms Always Exist and Can Be Computed!