The goal of this paper is to present an elementary, intersection-theoretical approach to the local invariants of plane curve singularities. We study in detail three invariants: the Milnor number , the number of double points  and the number r of branches of a local plane curve. The technique of Newton diagrams plays an important part in the paper. It is well-known that Newton transformations which arise in a natural way when applying the Newton algorithm provide a useful tool for calculating invariants of singularities. The formulae for the Milnor number in terms of Newton diagrams and Newton transformations presented in the paper grew out of our discussion on Eisenbud{Neumann diagrams. They have counterparts in toric geometry of plane curve singularities and in the case of two dimens 

The contents of the article are:

1. Plane local curves

2. The Milnor number: intersection theoretical approach

3. Newton diagrams and power series

4. Newton transformations and factorization of power series

5. Newton transformations, intersection multiplicity and the Milnor number

6. Nondegenerate singularities and equisingularityions imply theorems due to Kouchnirenko, Bernstein and Khovanski.

2000 Mathematics Subject Classification. Primary 32S55; Secondary 14H20.

The first author is partially supported by grants MTM 2010-21740-C02-01 and MTM 2010-21740-C02-02.

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