My boyfriend and I have been living together since 2015 and we had a baby this year, I have not worked since 2015, he handles all expenses. Can he claim both of us?

Yes. He can claim the baby since he/she is related (his biological child). He can claim you (an unrelated person) because you lived with him all year. There are additional rules, see below; but from your brief description of your situation, you probably meet them.

If he uses TurboTax(TT) to prepare his return, he should answer that the baby lived with him all year (this is allowed for newborns).

DEPENDENT RULES

There are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, a relationship test and a residence test. Only a QC qualifies the taxpayer for the Earned Income Credit, the Child Tax Credit. 

A child of a taxpayer can still be a “Qualifying Child” (QC) dependent, regardless of his/her income, if:

1. He is under age 19, or under 24 if a full time student for at least 5 months of the year, or is totally & permanently disabled

2. He did not provide more than 1/2 his own support. Scholarships are considered third party support and not as support provided by the student.

3. He lived with the parent  for more than half the year, or since he was born, for new borns.

A person can still be a Qualifying relative dependent, if not a Qualifying Child, if he meets the 6 tests for claiming a dependent:

1. Closely Related OR live with the taxpayer ALL year

2. His/her gross taxable income for the year must be less than $4,000 (2015)

3. The taxpayer must have provided more than 1/2 his support

In either case:

4. He must be a US citizen or resident of the US, Canada or Mexico

5. He must not file a joint return with his spouse or be claiming a dependent of his own

6. He must not be the qualifying child of another taxpayer