I currently have an H1B Visa.  My current employer/sponsor is considering moving his office to a different location, can I re-locate with the company under my current visa or will I need to file a new application?  Also, my employer may want me to frequently work overseas, may I do this with an H1B? 

You ask great questions and I am glad you are seeking help from an immigration attorney in advance of just acting without finding out your options.  Much of law is similar to preventative medicine and with careful planning, goals can often be accomplished.  With that, let's get started on your questions.
The answer to the first question of whether an H1B Visa holder is permitted to move with their employer is yes.  As always in law, there is a but.  The general rule is an H1B visa holder must work at the location listed on their Labor Condition Application (LCA) (part of the initial H1B application process).  If the employer wants you to work at a location other than the one listed with the LCA, the employer must notify the United States Citizenship and Immigration Service as well as file a new Labor Condition Application with the Department of Labor.  An exception to this would generally be if the new work place is within the area of intended employment listed on the Labor Condition Application, for example, an H1B employee's job duties require substantial or constant travel. 

The second question of whether an employee can perform work for an employer overseas is it depends.  For starters, there are no limits in the amount of time an H1B holder can spend outside of the United States while the H1B Visa is valid.  If you are travelling within the scope of duties listed on your H1B Visa application, then the USCIS may be less suspicious in whether you are still working for the H1B employer.  As mentioned above, you can travel outside of the United States for as long as your employer permits you without terminating your employment.  However, if you are not travelling  on business outside of the United States for an extended period  keep in mind that it may raise questions with the USCIS as well as Customs Border Patrol when re-entering the US in whether you are still working for the H1B employer.  

For starters, let's discuss what an H1-B work Visa is.  An H1B Visa, also referred to as the "Specialty Worker" Visa is a temporary visa made available to workers in occupations requiring highly specialized knowledge.  The Visa is commonly utilized by persons in fields such as computer science, medicine, law, engineering, business, and teaching, just to name a few.   There are two main components to an H1B, namely whether the applicant has the appropriate background and experience to be considered a specialty worker and whether the position requires the skills of a specialty worker.   For example, a compute programmer with a bachelors degree in computer science applying for a secretarial position will not likely be approved.  However, a computer programmer with a bachelors degree applying for a position whereby the job requires the applicant to develop software will generally be acceptable.  Person's applying for an H1B must have a job offer before applying for the Visa, as a sponsoring employer is required. 

H1B Visas are valid for a period of 3 years and may be renewed once for a total of six years.   Recipients of H1B Visas are generally permitted to bring their spouse and minor single children, however the accompanying spouse and children are not permitted to work unless they independently qualify for a work visa. 

Obtaining H1B Visas is very competitive as only 85,000 visas are available each year, 20,000 of which are exclusively reserved for people with a minimum Master's level degree from a United States Academic Institution.  H1B Visa applications are accepted beginning April 1 of each year for work to begin on October 1 of that same year.  Unfortunately, there are usually significantly more H1B applicants for available H1B Visas each year.  For instance, by April 3, 2007, over 119,000 H1B applications were received for 65,000 available visas.  As you can see, with the limited supply and large demand for H1B Visas, its important to put apply early and put your best case forward and is best to consult with an immigration lawyer.

Gerald Cipolla & Associates specialize in   Immigration Attorney Chicago
  and 
US Immigrant Visa Chicago  services.